OK, I know I shouldn't keep my PC in an enclosed space, but it makes the desk more tidy. If I can keep it from overheating, that's where I want to keep it. Right now I have to keep the door open whenever it's on. There is a vent in the back of the desk, but it doesn't seem to let much hot air out. Anyone have a suggestion for a fan to send hot air out of that vent? Should I try a large case fan (I've seen power adapters for the back panel), or is something a little larger available (opening is about 5 x 8 inches)? Will I still need to keep the front door open?

You will need to allow fresh ( Cool) air into the case.
So if the back doesn't have an opening to allow air in .
Then yes keep the front open.

Or make an opening in the lower section of the back panel to get air in .
Just make sure the air can flow to the bottom. front of case where the air intakes are