There are two broad reasons why the jump drives are only used on planets. The first reason is that some property which planets have but spaceships lack is required for the jump drive to work successfully. Two such properties come to mind: gravity and thermal mass.
The second reason is that it is possible to move smaller objects, but due to some scaling property of the jump drive technology it is not practical to move small things. This works best if we assume the jump drive to require a large amount of external infrastructure to operate.
In all scenarios, we require that the energy to jump scales slowly or not at all with the volume of space or amount of matter that is jumping. If this is not the case, then moving planets would be virtually impossible and the technology would instead focus on making spaceships suitable for the jump drives.
For this, we shall assert that the majority of the energy used in the jump drive is to initiate the connection between two distant points in space. The energy to envelope a larger region of space or transport additional mass does not grow too quickly, up to at least planetary scales.
You can use choose one or a combination of all of the below factors for your jump drives, adjusting their significance as you see fit.
Jump Drives Require Gravity Wells
Planets, by definition, have substantial mass and produce substantial gravity. Spaceships, on the other hand, are rarely large enough to have any noticeable gravity (except with very sensitive instruments); any exceptions to this would be plot fodder for your story.
By general relativity, gravity bends spacetime, with regions of high mass (such as around a planet) having high curvature. A wormhole requires bending space to the extreme. So let us postulate that the jump drive generates some sort of wormhole.
This jump drive requires existing curvature in spacetime (that is, an existing gravity well) to generate or contain a wormhole. Of course, small objects still have some gravity, so we have to apply some form of scaling. Either the amount of energy required to perform a jump goes up prohibitively quickly if there is too little gravity, or the distance of the jump is smaller with less gravity.
This scenario, where the jump drive converts an existing gravity well into a wormhole, has an interesting consequence. Compact objects such as neutron stars and black holes have extreme gravitational curvatures while often being only a few kilometres in diameter. This means that, if you could equip a neutron star or black hole with a jump drive, you could use it as a sort of super jump drive. I would posit that this would give the jump drive a ludicrous range, suitable for intergalactic travel. It may also be able to create a very large wormhole which could cover an entire solar system.
If a spaceship wants to jump independently of a planet, it would need to somehow generate a strong gravitational field. It might be able to do this with a miniature black hole, although the black hole can't be too small otherwise Hawking radiation would lead to its untimely evaporation. Creating and containing a black hole is a non-trivial exercise, though, so even if it is possible it won't be common.
Jump Drives Require Heat Sinks
Thermodynamics is a harsh mistress. As a consequence of the second law of thermodynamics, every process which does useful work also produces some amount of waste heat. The second law also dictates that you can't turn the waste heat into a more useful form (unless you produce even more waste heat to compensate). By the first law of thermodynamics, it is impossible to make that heat simply disappear. As such, you have to put that heat somewhere.
In space, the typical thing to do is radiate away that heat, using radiators and black-body radiation. However, there is a limit to how fast you can radiate your heat before your radiators have to be so hot that they'll melt.
The solution to temporarily having more heat than you can get rid of is to have a heat sink. This is something which can absorb a large amount of thermal energy while only rising a little bit in temperature. A good heat sink has a lot of mass.
The jump drives require an enormous amount of energy to be expended in a relatively short amount of time. This produces a stupendous amount of waste heat. So much waste heat, in fact, that any object as small as a spaceship would instantly vaporise.
Planets are far more massive than any practical spaceship, thus have a much larger heat sink than any spaceship could have. If you need something approaching a planetary mass of matter for a heat sink for your jump drives, then only planets can have jump drives.
Such a jump drive would have most of its infrastructure on the planet (although it can have power beamed to it from a local Dyson swarm), distributed across the entire planet as many nodes (the more the better). Each node of the jump drive would have 'roots' made of some material with superlative thermal conductivity stretching deep into the planet's crust and/or oceans. When the jump drive is operating, each node sends the heat it produces through its roots into the planet, increasing the planet's temperature by a tiny amount. Each node would also have a vast array of radiators ('leaves', perhaps) which it uses to get rid of that heat in between jumps.
This jump drive, which requires massive heat sinks, precludes their use with non-terrestrial objects (such as gas giants, stars or black holes) unless you build a megastructure with the mass of a terrestrial planet surrounding that object. This would not be practical for regular transport and trade, making it only useful if you need to move the gas giant/star/black hole somewhere else for whatever reason.
If someone wanted to use this jump drive with a spaceship, they would have to do one of two things. Either they would have to find a sufficiently massive heat sink (an asteroid, perhaps, or maybe some unobtainium bricks), or invent a vastly more efficient jump drive to reduce the amount of waste heat.
A consequence of this jump drive is the accumulation of heat if you jump too rapidly. While multiple jumps in rapid succession may be possible (recharging of capacitors/batteries permitting), that won't allow enough time to radiate away all the heat produced in the previous jump. To avoid jump drive induced global warming, you would probably need to have an average frequency of jumps much less than once per day (although this can involve multiple jumps in a couple of days followed by a few weeks or months of no jumping, although I would imagine general transit would prefer to space out the jumps).
Economy of Scale
The prior reasons are physical constraints which might be used to make jump drives impossible for small objects. However, we can instead invoke the economy of scale which can make it impractical to routinely transport small objects while still leaving open the possibility.
Let us suppose that the member solar systems each have Dyson swarms or some similar technology which harnesses an appreciable fraction of their star's power, and that this is the largest source of power in the solar system. This is expected for a Type III civilisation.
This jump drive requires an enormous amount of energy to function. So much energy that it requires most of the power of the Dyson swarm. As such, a given solar system can only operate a limited number of jump drives in a given period of time. This makes jumps per day a limited resource which must be managed carefully.
We require the scaling of energy with the size of the jump to be very small compared to the baseline energy needed to initiate the jump.
Because the number of jumps is limited, but the amount of stuff you can take per jump is not limited (or at least has a much looser limit), we want each jump to take as much stuff as possible. A planet probably has the most amount of civilisation-useful stuff in one place, so the optimal solution is to move the whole planet and everything on and around it.
Note that if this economy of scale is the only constraint, then equipping spaceships with jump drives is still possible. If the jump drive instead operates purely on external hardware (with a solar system spanning wormhole projector, perhaps), then any objects can be transported without having to equip them with a jump drive.
However, using a star system's power to operate a jump drive for a mere spaceship (or fleet of spaceships) would be the equivalent of shutting down a nation's airports, seaports and major roads for several hours. If a character or entity is able to get authorisation for such a move, then that character or entity (or their task) must be of galactic importance.
With this jump drive, if someone were to invent a compact and portable power source which rivalled that of a star and equipped a spaceship with that power source, then that spaceship could jump at will. But such a power source would be a game-changing technology.