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You can probably just buy a new atx power supply and drop them in. The standard hasn't really changed, just added new connectors and outputs.
If it's an AT power supply you'll need to find a used computer place. You can tell its AT if the power switch physically connects to the power supply (its actually switching the wall voltage) instead of the motherboard.
There's a good chance that it's the capacitors in the PSU that have failed. It's always the caps! Should be a very doable repair.
I was gonna write this^. Look for capacitors that look bloated at the top. Those should be replaced with equivalent ones.
2000's era
Any modern standard PSU will work provided it has molexs. PSUs were pretty much standardized by then, there's really no need to "rescue" the one it came with
Now if you had a "true" vintage computer with a non standard PSU, like something from the 80s or an Apple II or something like that, yea you might need to repair it or hope someone is making new replacements for it
Edit: Unless it's a damn proprietary PSU like Dell, pop the model in eBay and see what comes up in that case.
Well if it’s a prebuilt like a Dell a lot of those used non-standard connections
Ah right, except for that, damn propriety bullshit lmao
it's likely possible, but it's hard to imagine that it would be worth the trouble-- not to mention that it would also be pretty dangerous. replacing them would be far simpler.
If the computers have any value it would be better to just buy a new quality modern ATX power supply of the right size for the case (take dimensions for fit and ensure it's at least as many watts as the old one though it can be more) and do a drop-in replacement. Just make sure the power supply comes with some molex adapters and the maker usually sells additional cables if it's a semi or fully modular design so you could buy more molex plugs if the 3-8 they give you aren't enough.
That said power supplies can of course be repaired by anyone with soldering skills and sufficient electrical engineering knowledge and experience. They shouldn't be repaired by amateurs because they can store enough power to kill or maim a person who doesn't know what they're doing. It would be cheaper though to just replace the power supplies unless having all original/period equipment in the machines is important.