You can have your nine worlds sharing a single orbit around their star, as suggested in an answer by a4android. Or you could have your nine worlds having nine different orbits in their habitable zone around their star.
Nobody know what the absolute maximum number of planetary orbits within the circumstellar habitable zone of a star is.
It seems very probable that stars with no planets within their circumstellar habitable zones are many times as common as stars with one planet each within their circumstellar habitable zones, stars with one planet each within their circumstellar habitable zone are many times as common as stars with two planets each in their circumstellar habitable zones, stars with two planets each within their circumstellar habitable zone are many times as common as stars with three planets each in their circumstellar habitable zones, and so on, with systems with more planets within their habitable zones becoming rarer and rarer and rarer.
So even if it is possible for a star to have nine planets in its habitable zone that would be extremely rare; maybe one star out of a million, or one star out of a billion, or one star out of a trillion, or one out of some other vast number. The odds against any one specific star having nine planets in its circumstellar habitable zone would be, to use an apt term, astronomical.
But the universe is so vast, and the numbers of stars so astronomically vast (possibly infinite), that if it is possible for a star to have as many as nine planets in nine different orbits in its circumstellar habitable zone, there probably would be some stars with as many as nine planets each in nine separate orbits in their circumstellar habitable zones.
HOW MANY SEPARATE PLANETARY ORBITS CAN THERE BE IN THE HABITABLE ZONE OF A SINGLE STAR?
long ago, before any exoplanets were found orbiting other stars, Earth's solar system was the only known example of a solar system.
There was a time when scientists couldn't understand why the Sun and other stars emitted radiation and light. There is a story that at one scientific conference an angry astronomer punched a geologist, because the geologist insisted that geology proves that the Earth was many times older than astronomical and physical calculations showed was the maximum possible time that the Sun could shine for.
Between about 1920 and 1950, brilliant scientists calculated how stars produce energy by thermonuclear fusion within their super dense and super hot cores.
And people began to realize the implications for the possible habitability of exoplanets if they existed. For example, as I remember, Robert A.Heinlein mentioned that spectral class G stars would be the best for having habitable planets in juvenile science fiction novels like Starman Jones (1953) and Time for the Stars (1956).
As far as I know the first extended discussion of the suitability of various spectral classes of stars for having habitable planets was Stephen Dole's Habitable Planets for Man (1964, 2009).
On pages 49 to 52 Dole discusses the spacing of the planets in our solar system. He calculates that each planet's gravity interferes with the orbits of other bodies nearby, so that each planet has a forbidden zone where no other planet could form or have a stable orbit, based on the mass of the planet and the planet's distance from the Sun.
According to Dole's data, it might be possible to have more potentially habitable planets in the habitable zone of the Sun, orbiting in the habitable zone and in the spaces between the forbidden zones of Venus and Earth, or between the forbidden zones of Earth and Mars, or between the forbidden zones of Mars and Jupiter.
It is obviously perfectly possible for a solar system to form without any planets of the suitable size in the habitable zone, or to form with only one planet of suitable size in the habitable zone - like our solar system. But it seems to me that on rare occasions a star exactly like the Sun might have about 1.5 to 2.0 times as many planets of suitable size in it's habitable zone if the forbidden zones of those planets almost touch.
So depending on the size of the Sun's habitable zone, and on whether there are believed to be one, two, or three planets of suitable size in the Sun's habitable zone, a star exactly like the Sun could have zero, one, or two planets of suitable size within its habitable zone, and possibly as many as three, four, five, or even six.
No doubt solar systems with more planets of suitable size in a star's habitable zone will be more rare, so that each increase in the number of suitably sized planets in the habitable zone will have a significant decrease in the percentage of star systems having that number.
And of a course planets of a suitable size in the habitable zone of a star would not necessarily be habitable for either native life or for humans. Earth, for example, is of suitable size and in the habitable zone of the Sun, but did not develop a breathable for humans atmosphere until it was billions of years old.
Dole discusses the age of a planet that is habitable for humans on pages 61 to 63, and concludes:
In general, it is probably safe to say that a planet must have existed for 2 or 3 billion years, under fairly stable conditions of solar radiation, before it has matured enough to be habitable.
On pages 67 to 72 Dole discusses the characteristics of a planet's star. If a planet's star has to remain on the main sequence and emit a fairly steady amount of light for at least 3 billion years, it has to have no more than 1.43 times the mass of the Sun and a spectral Class F2 or later (Class F2 to F9, G Class stars, K class stars, M class stars). And to avoid having a planet become tidally locked to its star the star should have a mass at least 0.72 times the mass of the Sun, or spectral class K1.
So Dole concluded that stars with habitable planets in their habitable zones would probably range from class F2 to F9, G0 to G9, and K0 to K1.
And the more massive and brighter stars have more distant inner and outer limits of their habitable zones, and thus broader habitable zones. And the forbidden zones of planets would be relatively smaller in the broader habitable zones, making room for more potentially habitable planets possibly orbiting in the habitable zone.
So I long ago concluded that a solar system with the maximum possible number of planets in its habitable zone would probably be one with a spectral class F star as the primary that the planets orbit. So what is the widest possible circumstellar habitable zone?
The Wikipedia article Circumstellar habitable zone has a table listing various calculations of the inner or outer edge, or both, of the Sun's habitable zone.
And you will note that they vary widely in how broad or narrow they calculate the Sun's circumstellar habitable zone is. Some of the calculations may be for planets habitable for Humans, as Dole's were, and some may be for planets habitable for life but not for humans, which could explain some but not all of the radical differences.
The most commonly used definition of the inner and outer edges of the Sun's habitable zone is by Kasting et al. in 1993, with a conservative habitable zone from 0.95 to 1.37 Astronomical units (AU) and an optimistic habitable zone from 0.84 to 1.67 AU. So the outer edge of Kasting's conservative habitable zone is 1.442 times as distant as the inner edge, and the outer edge of Kasting's optimistic habitable zone is 1.988 times as distant as its inner edge. Kasting's conservative habitable zone is 0.42 AU wide and his optimistic habitable zone is 0.83 AU wide.
It is estimated that the habitable zone of a relatively hot F0 star would extend from about 2.0 AU to 3.7 AU and between 1.1 and 2.2 AU for a relatively cool F8 star.4
So a class F2 or F5 star could have a significantly broader circumstellar habitable zone than a class G2 star like the Sun.
HOW MANY SEPARATE PLANETARY ORBITS CAN THERE BE IN THE HABITABLE ZONE OF A DOUBLE STAR?
And long ago I decided that a binary star with two very similar class F stars orbiting together could have a wider circumstellar habitable zone than just one of them, with their planets orbiting around both of the stars instead of just one.
Since the intensity of a light source varies according to the square of the distance, an object 1.4142 units as far away from a light sources will receive half as much light as an object 1.0 units from the source. So if the light source becomes twice as bright, an object 1.4141 units from it will receive as much light as an object 1 unit from the source did previously.
So the circumstellar habitable zone around a closing orbiting pair of twin stars will be 1.4142 times as wide as the habitable zone around only one of them would be. So the habitable zone around a close pair of F0 stars would about 3.4142 to 5.2354 AU, 1.81834 AU wide, and one around a closely orbiting pair of F8 stars would be about 1.5556 to 3.1112 AU, 1.5556 AU wide. That is 3.7038 to 4.3293 times the width of Kasting's conservative habitable zone and 1.8742 to 2.1907 times the width of Kasting's optimistic habitable zone.
So depending on the maximum possible number of Earth sized planets that could possibly orbit in Kasting's conservative or optimistic habitable zones, perhaps two to three, and possibly more, a close pair of twin class F stars might have room for three to twelve, or possibly more, Earth sized planets each with its own separate orbit, in their combined circumstellar habitable zone.
And so for a long time I imagined that a close binary of F class stars would be the type of star system most likely to have the maximum possible number - whatever that number was - of suitably sized and potentially habitable planets in the circumstellar habitable zone.
HOW MANY SEPARATE PLANETARY ORBITS CAN THERE BE IN THE HABITABLE ZONE OF A TRIPLE OR QUADRUPLE STAR?
But suppose that the maximum possible number of separate planetary orbits in even the combined circumstellar habitable zone of two Class F stars would be only five. How could one get as many as nine separate planets in separate orbits in a single star system then?
By having four class F stars in the star system, with two binary stars each orbited by a number of planets, and the two binary stars orbiting each other at a much greater distance than the planets orbited. Then one could have five planets orbiting one pair of stars in their combined circumstellar habitable zone (plus other planets not in the zone), and four planets orbiting the other pair of stars in their combined circumstellar habitable zone (plus other planets not in the zone), for a total of nine planets in the habitable zones (plus other planets outside the zones).
But if a writer wants or needs for story reasons to have all nine planets orbit around one star or group of stars in the circumstellar habitable zone, having the nine planets orbit two different pairs of stars will not do. It would be necessary to bring all four stars, both pairs, close enough together that planets can orbit in the combined circumstellar habitable zone of all four stars.
If there are four F class stars of (almost) identical luminosity in the center of the system, their combined luminosity will be four times that of a single one of them, and thus the inner and outer edges of their combined circumstellar habitable zone will be twice as far as those of a single one of those stars would be.
Is is possible for planets to orbit in the combined circumstellar habitable zone of four stars?
Calculations show that in a binary (two star system) it is sometimes possible, depending on various factors, for planets to have stable orbits around one of the stars, called S-type orbits, or for planets to have stable orbits around both of the stars, called P-type or circumbinary orbits. In some binary systems it would even be possible for planets to have stable S-type orbits around each of the stars with other planets having stable P-type or circumbinary orbits much farther out around.
The first known planet with a P-type or circumbinary orbit, Kepler-16 b, orbits around Kepler-16 A & B, at a distance of about 0.704 AU, which is about 3.2 times their separation of about 0.22 AU.
On the other extreme, FW Tauri AB b orbits at a distance of about 150-300 AU from FW Tauri AB, which orbit each other at a distance of about 11 AU.
Multiple star systems are usually structured with at least one pair that orbit very closely. When there are two pairs of stars the distance between each pair is usually tens or hundreds of times the distance between the stars in either of the pairs.
Assuming that an F class star is about 1,000,000 miles or 1,609,344 kilometers in diameter, and each pair of F stars is separated by 5 times their diameter, 5,000,000 miles or 8,046,720 kilometers, and the two pairs of stars are separated by 5 times the separation of each pair, and so the two pairs would be 25,000,000 miles or 40,233,600 kilometers apart.
A separation of 25,000,000 miles or 40,233,600 kilometers would be about 0.268945004 of an AU.
since it is estimated that:
It is estimated that the habitable zone of a relatively hot F0 star would extend from about 2.0 AU to 3.7 AU and between 1.1 and 2.2 AU for a relatively cool F8 star.
And since the combined circumstellar habitable zone around a quadruple system of identical stars would twice the inner and outer radii of a single star of that magnitude, such a combined circumstellar habitable zone would have an inner radius between about 2.2 AU to 4.0 AU and an outer radius between about 4.4 AU and 7.4 AU, depending on the exact luminosities of the class F stars.
An inner edge of the combined habitable zone between about 2.2 to 4.0 AU would be about 8.1801 to 14.8729 times the separation of the two pairs of stars. So if the two pairs of stars are separated by a very short distance, close to the minimum possible I think, planets at the inner edges of their combined habitable zone should have stable orbits.
So a quadruple star system of spectral class F stars is the system that I suggest has the absolute maximum possible sized habitable zone while remaining reasonably plausible.
DO MORE MASSIVE OR LESS MASSIVE STARS HAVE MORE ROOM FOR PLANETARY ORBITS IN THEIR HABITABLE ZONES?
It may be noted that a small change in the mass of a star will create a much larger change in the star's luminosity. A star with twice the mass of the Sun will have more than twice the luminosity of the Sun. A star with half the mass of the Sun will have less than half the luminosity of the Sun.
The intensity of a star's gravity will fall off with the square of the distance. The intensity of a star's light received on a planet will also fall off with the square of the distance. But since the luminosity of the star will increase or decrease much faster than the mass of the star changes, planets orbiting in the habitable zone of low luminosity stars should be in much stronger gravitational fields than planets orbiting in the habitable zones of high luminosity stars.
And I am not sure which will reduce the width of a planet's forbidden zone more and make it possible for more planets to orbit in the habitable zone.
Would more massive and brighter stars have room for more stable planetary orbits in their habitable zones, or would less massive and dimmer stars have room for more stable planetary orbits in their habitable zones?
SOME RECORDS OF EXOPLANET SYSTEMS
By now, over 4,000 exoplanets have been discovered in other star systems, and sometimes two or more planets have been discovered in the same system. In some planetary systems planets are space similarly to the way they are in our system, but in other systems the planets discovered so far are spaced far more widely, or far more closely, than in our solar system.
DISTANCE BETWEEN PLANETARY ORBITS
In the Kepler-70 system, Kepler-70c orbits about 0.0016 AU or about 240,000 kilometers farther out than Kepler-70b.
During closest approach, Kepler-70c would appear 5 times the size of the Moon in Kepler-70b's sky.
The narrowest habitable zone for the Sun, calculated by Hart et al. in 1979, is between 0.95 AU and 1.01 AU from the Sun, or 0.06 AU wide. 0.0016 AU goes 37.5 times into 0.06 AU, so there is room for 38 planetary orbits separated by 0.0016 AU within Hart's habitability zone.
The much more widely used habitability zone calculated by Kasting et al. in 1993 has a conservative zone from 0.95 to 1.37 AU, 0.42 AU wide, and an optimistic zone from 0.84 to 1.67 AU, 0.83 AU thick. 0.0016 AU goes 262.5 times into Kasting's conservative zone and 518.75 times into Kasting's optimistic zone.
This questions requires that each planet has a natural satellite similar to the Moon. Since the Moon orbits Earth at an average distance of about 384.399 kilometers, the spacing between planetary orbits should be at least about five or ten times that much so that planets don't perturb each others's moons too much, and thus at least 1,942,995 or 3,843.990 kilometers - 0.012988119 to 0.02695486 AU.
So there should be room for about 3 to 5 separate planetary orbits with that spacing in Hart's zone about 0.06 AU wide, about 17 to 33 separate planetary orbits in Kasting's conservative habitable zone about 0.42 AU wide, and about 31 to 64 in Kasting's optimistic habitable zone about 0.83 AU wide.
RELATIVE SPACING OF PLANETARY ORBITS
The Kepler-70 system has the smallest spacing in kilometers or AU between planetary orbits. But what about the relative spacing of planetary orbits, the ratio between the semi-major axis of one planet's orbit and the semi-major axis of the next planet's orbit?
The Kepler-36 system has the smallest relative spacing of planetary orbits.
Kepler-36b and c have semi-major axes of 0.1153 AU and 0.1283 AU respectively, c is 11% further from star than b.
To be precise, according to the figures given in that list, Kepler-36c is about 0.013 AU farther out than Kepler-36b, and has a semi-major axis that is about 1.1127 times that of Kepler-36b.
Hart's habitable zone for the Sun has an outer edge that is only 1.0631 times as distant as the inner edge. Thus it could have only one planetary orbit in it, if the minimum relative spacing of planetary orbits is 1.1127 times.
Kasting's conservative habitable zone for the Sun has an outer edge that is 1.4421 times as distant as the inner edge, and Kasting's optimistic habitable zone for the Sun has an outer edge that is 1.9880 times as distant as the inner edge.
If the minimum relative spacing of planetary orbits is 1.1127 times, there could be four planetary orbits in Kasting's conservative habitable zone, and seven planetary orbits in Kasting's optimistic habitable zone.
If the minimum relative spacing of planetary orbits is 1.1127 times, the minimum ratio between a ninth planetary orbit and the first planetary orbit would be about 2.349766 - the ninth planetary orbit would have a semi-major axis at least 2.349766 times that of the innermost planet.
Note that absolute size of a star's habitable zone doesn't make any difference to the number of planetary orbits that could be within it if that number of orbits is determined by the relative spacing of the planetary orbits.
The only ways to find room for nine planetary orbits with the habitable zone of a star would be to use a minimum relative spacing less than 01.1127 and/or to have a wider relative habitable zone, with an outer edge that is relatively farther out compared to the inner edge, at least 2.349766 times.
The table in the Wikipedia article Circumstellar habitable zone lists a number of papers that calculate different inner and outer limits of the Sun's habitable zone. Thus various combinations could produce much narrower or wider habitable zones for the Sun.
the four innermost inner edges are give as 0.912, 0.87, 0.75, and 0.38 AU, while the four outermost outer edges are given as 1.70, 2.0, 2.4, and 10 AU. the various combinations of them give ratios of outer edges to inner edges of 1.8640, 1.9540, 2.2666, 4.4736, 2.1929, 2.2988, 2.6666, 5.2631, 2.6315, 2.7586, 3.2000, 6.3157, 10.9649, 11.4942, 13.3333, and 26.32157.
And it is possible that if the minimum spacing of planetary orbits is determined by relative orbital ratios and not by distance, the minimum orbital ratio will be less than 1.1127, and that planetary orbits can have smaller ratios than 1.1127.
It is also possible that the minimum spacing of planetary orbits is determined by distance, and thus many stars would habitable zones wide enough for nine separate planetary orbits and even for several times nine separate planetary orbits. So finding out how to calculate the minimum possible spacing of planetary orbits would be useful.
NEED FOR LARGE MOONS?
The original question desires that all nine Earth like planets in the habitable zone should have large moons. Unfortunately the best present theory for the origin of Earth's Moon is that it is a cosmic accident, the result of a planetary collision of a specific and probably rather rare type. So if the accepted theory of the Moon's formation is correct, all nine improbable Earth like planets in a star's habitable zone would also have experienced highly improbable collisions that created large moons for them, greatly multiplying the improbability.
This multiplication of the improbability would probably be equally vast if the nine Earth like planets share a single orbit or have nine separate orbits.
Suppose for example, that an Earth like planet has only a 0.10 probability of having a large moon. Then the probability that two Earth like planets would have large moons would be 0.01, the probability of three would be have 0.001, the probability of four would be 0.0001, the probability of five would be 0.00001,the probability of six would be 0.000001, the probability of seven would be 0.0000001, the probability of eight would be 0.00000001, the probability of nine would be 0.000000001.
Does an Earth like planet need to have a large moon for habitability? Nobody knows.
NINE PLANETS SHARING A SINGLE ORBIT
As some answers have stated, it is possible to have several planets, between seven and forty two, of equal mass and equally spaced, sharing a single planetary orbit, as suggested by a4android.
As a4android says, that is suggested in Sean Raymond's website, PlanetPlanet https://planetplanet.net/the-ultimate-solar-system/10 Which in turn is suggested by a paper by Smith and Lissauer (2010).
And such a planetary configuration might be impossible to happen naturally, in which case a highly advanced civilization would have had to have built the planets and/or moved them into their orbits sometime in the past. That civilization might have been one of the societies presently living on the nine worlds, or a civilization that was active millions or billions of years before the time of the story, depending on the needs of the story.
And some of the other orbital configurations I have suggested in my answer might also be so very improbable that a planetary system with those configurations might have to have been constructed by a highly advanced civilization sometime in the past.
Someone has calculated how many more or less Earth-like planets could orbit in separate orbits within the circumstellar habitable zone of a star. And the answer seems to be up to five, according to the sources.
However, the calculations were said to have been made for a red dwarf star. And the article doesn't say whether the maximum number would smaller, the same, or larger for more massive G type and F type stars.
Assuming the maximum number of stable planetary orbits with the habitable zone of any star would be five, any attempt to have nine habitable planets in one solar system would have to have at least two stars, or put all the planets equally spaced along one single planetary orbit in the habitable zone.