Is there a way to request categories for a game? The only thing I could think of doing was PM a moderator of a specific game, but I feel that could get overbearing for the mods.

Well you actually got it, ask the mods of the game you want new cateories for. However, keep in mind that new categories is a thing that needs to be discussed to see if it makes sense.

Thanks for the quick reply! I'm probably going to sound stupid for asking, but how to message a specific person on this site?

The PM system isn't implemented yet. You'll have to contact them through Twitter or Twitch for the best efficiency