We've discussed this a couple of times in the Discord, but how does the idea sound? We all get together and play something together, to me it sounds great. As for what we can play, there's a number of things. For full-on multiplayer, there's Town of Salem, with it's big-ass lobbies. For something 1v1, we could do a tournament bracket in something like Pokemon Showdown, or fittingly for P5 there's Shogi.
Are there any other suggestions? The only requirement is that it's either free-to-play or very cheap, and that if it's on PC it not be very intensive to run. Other than that, go nuts.