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If you don't want or can't afford ejecting matter, you are left with solar sail: since photons carry momentum, reflecting photons can allow you to harvest that momentum and use it to move.

Expand reflective leaves in a suitable way, and use the light coming from the star at your advantage.

Mind of two consequences:

1. If you reflect the photons, you are not using them to feed you energy. If you are absorbing the photon you get half of the available momentum.
2. Maneuvering can be complicated and quite time consuming

Lasers

Admittedly, they don't have to be coherent, just a directed light beam. You can gather energy from starlight and emit it opposite the direction you want to go. In principle it works just like any other drive, except, photons do not have rest mass, so you won't lose any of your precious materials in the asteroid. Yet photons still carry momentum so you will gain speed. If the laser it self is moveable, or you have multiple lasers, you can even steer.

Keep in mind though, that this is a terribly slow form of movement.

Ion thruster. https://www.nasa.gov/centers/glenn/about/fs21grc.html

Technically this is also venting matter, but "venting" to me implies large quantities.

Kinetic energy = mv². Energy increases as the square of the velocity. By ejecting small particles you minimize mass loss. By ejecting them very very fast you maximize kinetic energy.

A creature in space near the sun would have access to a variety of incoming charged particles from the solar wind; mostly protons but other things too. Photosynthesis uses radiant energy and so without some adjunct metabolic path these charged particles cannot be harvested for energy. Your creature captures these and then accelerates and re-emits them; an ion thruster.

Electrodynamic propulsion.

https://en.wikipedia.org/wiki/Electrodynamic_tether

Electrodynamic tethers (EDTs) are long conducting wires, such as one deployed from a tether satellite, which can operate on electromagnetic principles as generators, by converting their kinetic energy to electrical energy, or as motors, converting electrical energy to kinetic energy.[1] Electric potential is generated across a conductive tether by its motion through a planet's magnetic field.

This is contingent on the organism traversing an electromagnetic field. Some planets will have them; stars should have them. By creating an electrical field in a long "tail" the organism / spacecraft can push against an external field, turning energy into momentum.

Neither of these ideas is fiction; both are real. Although, of course not used by real creatures, as far as we know.

Low energy forms of propulsion would include Yarkowsky propulsion (basically, you heat the area opposite to the direction you want to go, as much as possible) and both electrostatic and electrodynamic propulsion and attitude adjustment. Solar wind sails are another possible way.

Your organism should be able to measure the electric and magnetic fields outside its meteoroid, and alter the distribution of surface charges (or maybe even expel charged particles) as well as inducing currents within itself. Current control alone can be used for attitude control (the meteoroid aligns itself by exploiting the available solar magnetic field).

To quickly rotate around its barycenter, your organism would have to first excavate a spherical chamber all around said barycenter, then fill it with a solid sphere of the densest material it has at its disposal. Finally, fill the gap between sphere and chamber (just inches thick) with cilia. By moving the cilia, the organism can rotate the sphere any way it wants - and the outer meteoroid will rotate in the opposite direction, proportionally to the difference between the masses (if the sphere is 1/100th of the meteoroid mass, then by rotating the sphere 100 times the meteoroid will spin on its axis once).

Your being could utilize the Interplanetary Transport Network (ITN), which is a collection of gravitationally determined pathways through the Solar System that utilize low-energy transfer to minimize fuel use.

Low-energy transfer trajectories are paths in space where orbit around one mass is exchanged for orbit around another at points in spacetime where very little energy is required to effect this change. Longer trajectories can be combined from several such transfers; a method used by deep-space probes. While it would use little energy, transport along the network could take a long time.

Fuel for the transfers could be gained by collecting interplanetary ice or dust and accelerate it to relativistic velocities by linear accelerators powered by collected sunlight or fusion reactor.

It depends on whether you're willing to go into the realm of sci-fi, or you want to stay with the pure unadulterated science.

If you want something realistic, then the best options were already mentioned previously by people here: solar sail, photon or, ion drives(they use thrust, but minimize matter used).

However, if you're willing to go for something more... controversial, I remember that there are a few thrustless possibilities related to Mach's principle, that are most likely incorrect, but could be used for a sci-fi. It's been quite some time, and I only remember rough principle, but here's wikipedia article that you can use as basis for what to look for: https://en.wikipedia.org/wiki/Woodward_effect.

Capture the solar wind and use it to power an ion drive.

This will be very-low thrust, but it is feasible. Sources vary, but the density of the solar wind at Earth orbit is about 4 particles per cubic cm (electrons, protons and alpha particles).

You could have the organism capture the solar wind continuously, and only use it for thrusting occasionally, thus allowing higher thrust when maneuvering is required.