You can look up part specifications on the board schematic. You can download a (not very good) scanned copy of the manual here: https://www.ipdb.org/machine.cgi?id=1176. Page 56 has the board diagram with parts list. Seems to me that, at a minimum, you'd want to replace everything that's burned out: the resistor array, the resistors and capacitors, and possibly the transistor immediately below them (hard to tell if the problem started there, or there's just smoke that got on it).

Of course, you will want to try to figure out what caused the problem in the first place. If you just accidentally shorted something and know it's not going to happen again, that's fine, but if these just blew up for an undetermined reason, just replacing those components is probably just going to lead to them doing that again. In either case, probably want to buy extras.

As for source, any of the usual electronics supply places should have what you need. I'm partial to Digi-key, sometimes order from Mouser, and have heard that Great Plains Electronics and Newark Electronics are also good stores but have never used them myself.

And yes, it goes without saying, make sure you have the skills to tackle this kind of work. That said, if you screw it up, you most likely still have the option of sending it to someone more qualified; you're unlikely to completely destroy the board, even if you do make things a little worse it should be fixable, though at more expense. Also, used boards of unknown quality can sometimes be found on eBay, Craigslist, etc. and if all else fails you can buy a brand new replacement board, like this $500 option: https://www.pinballlife.com/rottendog-williams-system-9-11-11c-mpu-board.html