Geeze... is it final project time already?
Using an RC car chassis (and electronics) is an easy way to get the robot moving. All you really have to do is figure out how to move the paintball gun and how to control it.
You'll probably want a couple servos to control the mount the gun sits on. How big and how powerful I don't know, that all depends on how much force is required to move the gun and how quickly you want to move it.
How do you plan on aiming this gun? By watching the bot? That'll probably be a bit hard (as you won't always be behind the gun).
I believe (don't quote me here) that you can use an old PS2 controller with webbot lib and an axon. Someone was even talking about getting a PS3 controller with bluetooth working. Do you plan on having a wired controller, or wireless?