I've finally figured out all the networking issues with Gamemaker, and have an internet server running. I'm using a tool called GameMaker Server, which is really just a server a guy provides free to use, along with some built-in networking functions, like accounts, high scores, and achievements.
I don't really have much of a 'game' at this point, everybody can connect and their sprite is a random color, so players don't get confused. I'm still thinking of exactly what kind of game I'd like to make. I was planning on just making a 2 player capture the flag, and I think I'm still going to do that.
More updates to come as I continue working.