All software can have vulnerabilities, and video games are no different. It is certainly possible for games to have vulnerabilities, especially multiplayer ones where players may be in direct communication with each other.
Many multiplayer games are peer-to-peer (P2P), so players exchange packets with each other directly. In this case, there is no server to validate the traffic before it is passed to other users, so malformed packets could potentially crash the game, or worse, result in code execution. Of course, you'd hope that game developers would follow best practices and avoid these issues, but games are very complex and often still have many bugs.
I'm not saying server-side games are immune either, but are probably harder to find/exploit without getting caught.
It seems most people who look for vulnerabilities in games are just looking to cheat; getting free in-game money and becoming invincible etc. But there definitely could be issues that allow compromise of other players' computers; although I'm not aware of any.