So we recently bought a house last week (not new construction, 7 years old). Anyways, the return vent for one of the AC units (I think probably the largest one but not positive), is in a game room off the kitchen area. I realized that when the system is running it creates pretty significant negative pressure in that room. I can barely shut the door.

I probably need to have an AC company come take a look but should it be as simple as just move that return to the other side of the wall so it's accessing the whole house and not confined to that room? Especially because we plan on replace that door with a normal door (young kids) that would seal even better.

By the other side of that door, do you mean in the kitchen? Typically, you want the return air in a central area such as a hallway. I would be careful about it too near the kitchen as it can compete with the exhaust from the kitchen vent. It also sounds like it needs to be moved to a more central area.

I have this problem with my master bedroom and its the furthest point from the downstairs return, which is centrally located in an open area. I chalk it up to an older home and needing another return.

An HVAC contractor may be able to rebalance things with some dampers or ductwork changes. Other options would be to undercut the door or add a jumper duct or door vent.

Better idea is to simply add another return in the kitchen from the same duct. That will help equalize the pressure and you will pull air out of both rooms to draw new air in.

Or you can add a transfer above the door.