I've got an old Pentium III box, which still has some life in it. However, I need to replace the PSU. The current motherboard is an Intel (Fedora) model, with the 82815 chipset. Can I use a new, off-the-shelf ATX PSU with this board? According to the specs, the current (dead) PSU is a "200-Watt 3.3-Volt LP Power Supply with 700ma."
You should be able to. Just make sure it supplies enough +3.3V and +5V current for the board; modern MBs, CPUs and graphics cards demand lots of +12V power, so modern PSs often have lower +3.3V and +5V ratings than older PSs with similar total wattage ratings.
Your old PS may have a sticker on the side (as almost all new ones do) indicating how much current it can supply at the different voltages.