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With respect to its biological functions, Murderbot says in All Systems Red that it connects its resupply and repair leads, it says it sweats and breathes in several books and it says that it doesn't have a digestive system. That's about it. Martha Wells has also said that it doesn't heal on its own, and I at least don't recall seeing anything that would contradict (or obviously support, for that matter) that statement in the published books (up to System Collapse). It also says it doesn't sleep but has to take recharge cycles sometimes (during which it's unconscious).
The recharge cycle… recharges what exactly? Murderbot doesn't say, and it's unclear where the energy of recharging comes from since it doesn't plug into anything for recharging. One speculation is that it has a small, slow cold fusion power reactor inside it and that slowly provides its other batteries with power. And the other, faster batteries are what power its inorganics while it's up and running.
What do the resupply leads supply? Murderbot never says.
While Murderbot doesn't seem to sweat that easily, seeing as it can just sprint and run around without sweating enough for it to mention that, it can sweat a lot. Especially in that one particular scene in Network effect. Where is that water coming from? Is it one of the substances that is resupplied? Granted, it doesn't tell when it exactly has been resupplied (in fact, it hasn't mentioned that at all since the first time in ASR), and it might not know that it could run out of water. Or maybe it can't. If water isn't resupplied to it or the amount is too small, it is possible that it could have a device inside it that extracts water from air, maybe the air it breathes.
Murderbot has also human organic components, at least its skin, lung(s), (some of its) brain and to some extent its muscles. Regular human cells are continuously dying and dividing, being replaced, so it is difficult to see how Murderbot's cells wouldn't also do this kind of self-regeneration and thus self-heal to some degree. Perhaps future genetic engineering has managed to turn that feature off in constructs' cells or slow it down enough that it looks like constructs don't heal on their own.
Murderbot says its veins are self-sealing (though that feature fails sometimes), which implies that organic components are being manipulated with inorganic means. (Or possibly the veins are wholly inorganic, though I have no idea how they would integrate with the cells then.) But what does it mean for a vein to have sealed? Is the blood or other fluid flow permanently cut off until repairs can be conducted and is the organic component at the end of the vein out of luck? Maybe Murderbot has superflous routes so that cutting and sealing one vein will only have a small effect. Or the sealing means that the vein is actually reconnected to some extent and will then continue transporting blood or another fluid. The sealing mechanism would need to be quite sophisticated for that, but considering all the other technology, it is quite possible. This option is also vaguely supported by how Murderbot said in Network Effect that its "organic nerves didn't want to reconnect" when it severed and reattached its hand. With organic nerves, it would be reasonable to expect that once severed, they will obviously not reconnect until they're repaired, but Murderbot went out of its way to explain, which implies that reconnecting them would be possible in the current situation.
Regular human cells use oxygen and sugar and some other nutrients as they live and do their thing. If we assume that construct cells don't continuously divide and die (heal) or do so very slowly, they probably don't use as much nutrients as human cells. But they would still use some, especially muscles when they contract, and brains when they think. Murderbot does breathe, so the cells do get oxygen, but where do the sugars come from? They could also be another substance that is supplied via resupply leads. It is also a possibility that Murderbot has a tiny synthesiser (or recycler!) inside it that synthesises sugars and other nutrients when needed.
What are the other fluids besides blood that constructs have and Murderbot complains that leak out of it when it's been hurt? Could be anything. The above-mentioned nutrients, coolant, lubricant, nanobot slurry, just plain water…
How exactly do the repair leads repair Murderbot? Maybe I'll speculate another time.