In the Next Generation episode Silicon Avatar there is a crystalline entity that the Enterprise chases after. The Enterprise chases this entity at warp speed. The entity, however, is just a crystal structure. How was it that this crystal structure was able to achieve warp speed without a power source?
The Crystalline Entity is not some ordinary biological organism. As it stands it would barely be life as we know it. Though the Entity was destroyed, it did attempt to establish communication with the Enterprise crew (using graviton pulses) so its level of technical sophistication is also unknown. Since the Enterprise never did get to study the Entity fully, it could have been an amalgam of technology and biology such as the theorized Von Neumann probes speculated by scientists who believe future intelligences will create such probes to cross and study the vast distances of space.
What we know about the Crystalline Entity:
The entity is enormously powerful. It was able to be detected by the Enterprise sensor while in a planetary atmosphere at a distance of nearly a light year. At warp 8, Melona Four was still 6 HOURS away. (.7 light-years distance)
It converts matter to energy, seeming to prefer organic matter, and able to strip a planet of all its organic material in a matter of hours. We are not sure if it used any material besides organics but in the scene where the Enterprise crew leave the cave, there were deep and terrible gouges in the surface of the planet. Perhaps the creature uses mineral resources as well.
Its dietary habits were also quite strange considering it consumed all organic matter, trees, grass, insects and even bacteria from the soil and water. Given this was a planet of at least Earth size, that alone would be a considerable amount of energy if converted in a perfect conversion from matter to antimatter.
These two factors alone indicate a creature capable of generating incredible energies at least as powerful as a warp core. We learn the creature emits anti-protons both as it feeds and as it travels, this also makes sense if the creature is capable of warp travel since it is a common practice of Alpha Quadrant races to use matter-antimatter reactions to power their warp field generation.
It could be speculated the Crystalline Entity was not an organic being but a highly advanced machine. As such, warp space folding could be within its capabilities as a space probe. The death of the human colonists and planets where the device fed may simply not recognize US as life as THEY (its creators) knew it.
If it is nothing more than a naturally occurring form of life, then the universe of the Alpha Quadrant has even greater diversity than the Federation currently understood of life.
There's no canon in Star Trek that a biological entity can not travel at warp. Beyond that, though, we have very little data on the Crystalline Entity. Even the Memory Alpha entry on the Crystalline Entity has very little information on the nature of the being.
The entity appeared to be able to feed almost directly off of life forms, perhaps by absorbing electromagnetic energy directly, which indicates that it is not using something as inefficient as, say, a human body, to power itself. It would absorb the energy directly without the need to convert from food to energy, which would provide it with more energy than if it just "ate" matter.
It also detected and was able to admit graviton pulses, indicating, again, that it was operating in a different fashion than any known biological life form. It might not have been any form of LAWKI (Life As We Know It).
Also, as a side note, it would almost have to travel at warp speed to survive, unless it could stay dormant for extreme periods of time, since it fed on life, it would have to travel from star system to star system.
Also, since we know very little about the entity, it could have some connection with subspace or might even exist more in subspace than in normal space, so it could be tapping energy that isn't in this universe. But we do know it did have a power source - since it did absorb energy and it also used a "beam" when absorbing energy, so it worked with power and energy differently than most organisms we know.
There's nothing to tell us just how it did travel at warp speed, but there's plenty of evidence that shows it absorbs and uses energy more like a machine or computer than like a biological unit. It could even be the creation of some unknown species. But we do know that with it working with absorbing and using energy, that it did consume and use energy, so it did have power. The original source would be the life it consumed, but after that, the source would be itself, with the energy stored within.
In a very simple statement of actual physics laws: E2=m2c4+p2c2 (Correct version of e=mc2)
The energy output^2 is equal to mass^2 times the speed of light^4 plus the momentum^2 times the speed of light. Or simply put, the momentum + mass of an object equals the energy output. There are approximately 10x6.69^10 electrons in a single AA battery. And in the area of 10x1^250 atoms of oxygen in out atmosphere. If the Crystal was even a fraction of of the size of the atmosphere in terms of density, it would be able to travel for thousands and thousands of years at the speed of light if it had an equivelant mass.