Yes, this is possible; to make it easy on yourself, I would stick with a "standard" hobby R/C vehicle for the platform, as they are controlled via servos and/or ESC (electronic speed controllers), which are very easy to interface to most microcontrollers. Unfortunately, such platforms aren't cheap, but they are very rugged and powerful, which might be needed for your application (ie, you might want to add some "armor" to them so that certain parts aren't "pegged" with paintballs, gumming up the works (which you would need to clean and/or replace - and parts can get expensive, too). The nice thing about such a platform, though, is that if something breaks, it is fairly easy to fix (because the parts will likely be available).
If you have some mechanical skills, you might be able to build your own "tank"-style platform, similar to the r/c paintball tanks here:
https://rctankcombat.com/
It wouldn't have to be as "advanced" - you could, for instance, use simple inverted (turned "inside out") timing belts for the "treads", and custom sheaves for the wheels/bogies - on a small body, etc. The motors you could buy surplus, and buy or build an h-bridge to drive them (purchasing such a h-bridge might be preferable, as many have R/C servo inputs).
Another possibility would be to use a modified child's ride-on toy (like a Powerwheels ride-on).
For the on-board microcontroller (or, depending on the size of the platform, you might be able to use a larger PC motherboard), you would need some kind of wireless communication (wifi ethernet, or RF communication modules, or xbee) for reprogramming, etc while out in the field...