Their resources most likely include nebular gas and dust, or perhaps asteroids, given their lack of planetary occupation. The only interest they ever take in planets is for destruction (vasuda prime).
As to food, (assuming they are organic) they may have some way of metabolizing compounds found in nebular gas or asteroids (ammonia and methane ice) for energy or whatever molecules they require. However, given their almost metallic exoskeleton, power blades and plasma cannon (hallfight cutscene), it's not a huge stretch of the mind to imagine that they are not organic.
They could be some sort of robot, responding only to a higher intelligence. (hive mind robots). Should they not be organic creatures, then all they would require is energy and spare parts (i suppose), and seeing as to the low value they place on shivan individual life, they might not even bother with the latter.
This would bring on two questions: were they created, or did they evolve?
Let's say they evolved. Consider: humans have already created mechanical hearts, and artificial bones and joints. Soon we will be able to grow our own skin. With a dialysis machine we can replace the function of our kidneys
Granted, none of that works very well right now, but that's beside the point. We can already self-enhance by enriching our blood to carry more oxygen (done with cyclists sometimes), and gene therapy allows us to fix some problems, at least in a limited fashion.
Who says that genetic, biomedical, and techonological advances couldn't make us superbeings like the shivans? Perhaps after millions or billions of years of evolution, we might have decided that such an evolution (maybe ONLY that evolution) of the human race would best ensure our survival. Any other subspace-capable races would have to be eliminated as a matter of protection.
This is all of course, off the cuff.
If they were created, that would easily explain away all the strange morphologies and behavior (xenocidal, inorganic and organic melds). However, why were there two different waves of shivans, with such different goals and behavior? The first shivans sought to actively destroy our fleets, and they also effected a planetary bombardment on vasuda, an action the second wave of shivans never repeated. We can maybe assume that the creators are long defunct and the shivans went separate ways, but then why such a similarity in technology?
Also, are they intelligent (on the level of humans)? It seems more likely that they are a hive mind, but might they be a collective intelligence? in FS1 this seems much less likely (the destruction of the lucifer), but in FS2 it might make sense. They throw cruiser after destroyer after corvette after fighter after bomber without regard to losses, or even any clear strategic objective. Their only objectives seem to be "Kill it. If it's bigger, that goes first." For example, 80 juggernauts are left to blow up the capella star, something they could do without a care in the world were the system occupied, yet battle is in full tilt between the allied remnant and the shivan fleets. A juggernaut at each battle location would have completely wiped out all resistance to the shivan assault, yet the only one that ever sortied was the one that destroyed the colossus. What about the Aquitaine? It was attacked twice with the intent of destruction in the nebula, yet in later phases of the war it appears to have been ignored.
personally, I offer no explanation for the shivans. Their evolution was as random as ours, or the vasudan's, or the ancient's. I'll wait for the third installment

(oops, that's taboo around here)