After initially struggling to wrap my head around networking in Gamemaker, I eventually found a tool that helps do most of the networking for you. It's called GMNet Engine and it works by having the master game server running on a platform like Digital Ocean. The documentation for the tool was pretty extensive and I worked through the tutorial pretty quickly to get it up and running. The only thing I need is the Digital Ocean server, or to see if I can run a Java server on my A Small Orange hosting.
A little clarification of my game concept too. It's capture the flag type game where players have to retrieve an item from the middle of the screen and return it to their base. There will be obstacles and hazards to make retrieving the item a bit less trivial, and players will have a special attack that forces the opposing player to drop the ball.
More updates to come.