Secondly,
PIII does not need 5V heavy at all, In early days was common to see 200 to 300W loosely based on baby AT PSU design adapted for ATX standard.
Really, you don't need to worry much, get a new gold or bronze 80 Seasonic 550W with PFC PSU and it does have 20 pin plug which I use to test any ATX boards that are Pentium, PIII slot 1 and socket 370. As long as you don't have a card that needs -5V which is ISA audio cards.
Also I have older but still good Seasonic 380W waiting to be used. It was in other computers for a time in our household.
Only and very specific cases for two things:
If you have a sound card ISA with -5V pin on the connector (look it up on wiki to check against yours, even there is a gold finger make certain it is connected with a track or not?) and secondly especially Athlon slot A and Athlon XP socket 462 needed very fat current capacity 5V (over 30A). This is not needed if and especially if the socket 462 motherboard does have CPU 4 pin 12V plug next to the socket like any modern motherboard does.
No ISA slots motherboards then no need for -5V. If odd board still needs -5V, that is very rare as this usually for high quality serial ports.
Cheers,
Great Northern aka Canada.