TL;DR: Yes, it's possible
Summary: Yes such a planet is possible. The main part that is required is plant-like extensions that perform photosynthesis. But otherwise it works, without gravity an organism can grow to a large enough size to hold an atmosphere.
So first we need to get a list of what life requires in a planet:
All planets with life need to have some sort of energy source. The most common one is a star, but probably there are other options. The energy source needs to be capable of bringing the planet up to the proper temperatures for liquid water (see below).
This one is quoted a lot because it's rare. It's also integral to almost all life on Earth. For a planet to have liquid water, it has to have a certain range of temperatures (namely somewhere between 0 and 100 C) on a significant portion of the planet. The range a planet must be from its star to have liquid water is called the Habitable Zone. So for a planet must be consistently within the Habitable Zone for it to be inhabited (makes sense, no). There also needs to be water on the planet in the first place.
A planet generally needs a certain mass to sustain life. There are two main reasons for this. First, a larger mass more easily holds an atmosphere, which is necessary for earth-life. Second, a small planet generally doesn't have much geological activity. Small planets have a small diameter, and they lose most of the extra energy obtained from formation very quickly. This loss of energy results in a lack of geological activity. Though, there can be other options to produce this geological activity. For example Jupiter's moon Io has lots of geological activity because of the energy it produces orbiting Jupiter.
Another side effect of having a larger mass, is that most larger mass planets have an iron core. This iron core allows the planets to produce a magnetic field, which protects the planet from stellar wind and cosmic radiation. Mass isn't the only thing that makes a magnetic field, but it does effect the process.
Orbit and Rotation
For a planet to support life it has to have a "reasonable" orbit. If the planet swings too far out from its star, the its temperature will drop to far to support life. If the planet swings too close, all life would be killed by heat. Even the planet changing the length of its orbit but still staying in the habitable zone could be dangerous for life, unless the change was slow or predictable. Life can adapt, but only adapt so fast.
Rotation also plays a big effect. Rotation controls night and day cycles, and needs to be balanced in order to prevent the planet from heating up/cooling down too fast. The planet also needs to rotate on a tilted axis. First, this produces seasons, causing biological diversity. But a tilted axis also spreads weather around, helping to regulate the temperature of the planet.
Other factors in what in a planet needs to be habitable include the right elements on the planet, finding these elements in the right places, and those elements being released at the right time. The most common elements in life are carbon, oxygen, hydrogen and nitrogen. These are also some of the most common elements on Earth, and this helps make life possible.
It is also important to remember that organisms can live in strange environments. Organisms can flourish in places without oxygen, under high pressures, and in volcanic bits of earth in the poles. And that's just on Earth.
For a full description on the factors of a habitable planet, see the Wikipedia article.
Now for How the Organism Works
First, I'll address what the organism needs to be a habitable "body."
- Energy Source: That's easy, the planet-organism would be orbiting a star.
- Liquid Water: Likely these planet-organisms would be imparted with a large amount of water at birth. Their parents would likely harvest it from nearby planets with liquid water on them. The planet-organism probably would have a means of propulsion, so it could keep itself in the habitable zone
- Mass: These things can grow to any size you want them too, their is no constraint of gravity. As for how the planet-organism could stand its own gravity, it would either be need to pretty low density, or have very strong supports (see Final Notes section). This question does have some information, but not a lot. A magnetic field is harder, but who's to say the organism couldn't have a magnetic in it's body (see Homeostasis)?
- Orbit and Rotation: The organism could easily control its own orbit and rotation if it has the ability to move (some means of propulsion).
- Finally the organism would likely be born with the right amount of chemicals for supporting life. Adult organisms would gain these chemicals by eating asteroids or eating them off existing planets.
I haven't focused a lot on how the organism fit a habitable planet because you can make anything you want. The problem is fitting a habitable planet and fitting the constraints of life. So on to how such an organism could live under the constraints of life.
All organisms need to maintain a stable internal environment. Such a large organism would need a lot of power to keep itself heated and going. Fortunately, it would be orbiting a star, some of the ultimate source of energy. The plants on it would provide much of its energy, and possibly some energy would be absorbed directly through the skin. The also needs to be insulation to keep it warm. The way this organism would do that is to have an atmosphere. Similar to how earth maintains livable temperatures, this planet-organism would use gases to keep it and its surface warm. The poles would be cold, but likely the organism would have extra insulation there (insulation under the skin, like fat, not extra atmosphere).
The biggest problem with maintaining homeostasis is the organism having a magnetic field. One option would be having the organism have a large generator in it's center, producing a magnetic field. Such a generator would consist of two magnetic somehow revolving around each other. This revolving could be caused by the rotation of the planet. Another option is having the organism have an iron core. This is less likely, because the iron core needs to be liquid, and very hot, and the organism also has to obtain large amounts of iron. The final possibility for a magnetic field replacement is that the organism has some ability to repel solar winds, outside of a magnetic field. This method would be some sort of special organ, or possible a special element in the atmosphere. But by far the best option would be for the organism to have a similar ability to the electric eel. This organism would be able to control electricity and have it move around and around it its body, forming an electro-magnetic field. This answer has a few more details.
Living organisms need to be organized into cells, tissues, organs, and organ systems. This isn't hard to imagine even in a massive organism like this one. There would just be a lot more of everything.
This organism would likely live off of itself. Animals living on its surface would produce the necessary gases in the atmosphere for it to metabolize. As the question states, the actual energy would come from "plant-like" substances that are actually part of the creature itself. But these plants could perform photosynthesis.
One problem is how the planet could obtain materials besides the common nitrogen, oxygen, carbon-dioxide, and hydrogen. Likely this organism would be engineered in such a way that it needs minimal amounts of other elements. But it's main source of minerals would likely be asteroids, comets and other wayward cosmic bodies. When these objects come into the planets line of orbit, it ingests them and uses their minerals. Another option is that there is another space organism that exists in symbiosis with the planet organism. This organism needs the atmosphere of the planet organism to metabolize. But it doesn't have life on it, hence it can leave the habitable zone for relatively short periods of time. During this time it can consume bits of planets and asteroids, bringing back materials such as water and minerals. Anything the symbiotic organism doesn't need, is ejected onto the planet organism via waste. Then the planet-organism can absorb the minerals through the plants on the surface.
Growth, Adaption, and Response to Stimuli
All organisms need to be able to grow and develop, adapt to the environment, and respond to stimuli. I'm not going to spend much time on these, because these can be configured however you want them to be. But in brief:
The organism would be born small. It would grow and develop over time, likely relying on the parent for basic material and atmosphere (see Reproduction). The planet-organism would be able to adapt to its environment. Along with responding to stimuli, this would mean things like moving its position to stay in the habitable zone, changing how it behaves depending on what organisms are on it, and other various things. These three features could easily be handled within the parameters of such an organism.
Reproduction may be the hardest thing about these organisms to make realistic. First, I want to talk about sexual reproduction. The major problem with sexual reproduction is that it requires two of the organisms. These organisms would generally be a danger to each other because they would throw each other out of orbit. To be orbiting the same star as another organism would require more energy because they organism would need to adjust it's obit at each pass. If these organisms did reproduce sexually, I'm guessing they would stick together in pairs, orbiting the same star (though not necessarily next to each other). Likely the actual transfer of gametes would occur outside of the organisms bodies (i.e. in a tube in space), because they are two big to get really close.
Another possibility is asexual reproduction. If this is the case, the organism would immediately start with a zygote, which would start growing inside a womb-like area of the organism. At a certain point, the baby would leave its parent and exit in to space. At this time it would be very small. It would likely live off of its parent while it grew. Possibly it would be able to leave the habitable zone for short times to harvest minerals and water from nearby planets and asteroids.
When the baby had grown to a large enough size, it would separate from its parents, taking some of their organisms with it. See below for more info.
When these organisms left a star, their plant like extensions would be come useless so would be killed. They would die on the planet. But the organism needs animals to produce carbon-dioxide so it can photosynthesize. So likely it would have some special place to harbor animals while it passed through space. This place would produce glucose and oxygen for the animals to survive until they could be released to the surface.
A baby organism would receive animals from its parents. Likely the organism would lay on the surface of its parent, with the special cavity shown above open. When many organisms wandered into the place, which would be a good place to live, then the cavity would close and the baby would be on its way.
When the organism traveled through space, it would hibernate, using as little energy as possible, because it needs to store energy to be able to grow planets once it reaches a suitable star. As it travels through space, it may be that the planet-organism hits asteroids and comets. This would be stuck to the organism's surface, and could be absorbed when it pulls out of hibernation. So baby organism (when fully grown), could get some minerals and water this way.
The plant like structures of the organism would produce fruit, though not for reproduction. This fruit would solely sustain animals on the planet's surface. As I said before, the animals are necessary to produce carbon-dioxide for photosynthesis. This is another symbiotic relationship.
As for anatomy, much of the organs and things would be in the center of the planet. The surface would be a rocky, dead "skin." Possible in the center, where gravity is highest, there could be some sort of "furnace" for extra energy, but I think this unlikely.
For the organism to support itself, it is going to need very strong "bones." It might need to have a dead or hollow center, possible filled with iron (to help generate a magnetic field). The bones would be frequent and very large, all cells would attach to the bones, to provide support. The bones could also serve as a circulation system, getting smaller as they go deeper into tissue. If the circulation system was providing significant support for the entire organism, maybe it could better support it self. Of course there would need to be someway for the materials to leave the "blood" (or phloem), so the bones would have to be very porus.