An old Beetle will certainly do the job for that. I wouldn't worry too much about reduction boxes, as you should be able to get wherever you want to go in first and second gear. I'm assuming that you will just use any old cheap tyres and won't need to bother with proper off-road tyres? What kind of budget are you looking at for the whole thing? If you want good grip and added ground clearance then bigger tyres are the first step. If you're on a budget then this may require getting some used 4wd tyres and then perhaps converting your stud pattern to fit the big rims using adapter plates. Adapters are easy to make and will work for your application although they're not road legal.
Automatic is kind of an issue, because it won't be particularly cheap or easy to find one to suit a bug. Is it important to have an automatic transmission? If the driver is focused only on driving and not fighting, then they should be able to handle a manual transmission easily.
One question that I would ask is, how big does it need to be and how good does it need to be off-road? I'm assuming that this will be used only on paintball fields, so they should be mostly flat paddocks and not too many big obstacles? Maybe a bit of muddy forest here and there?
If I was going to build a vehicle like you describe, I would buy any old automatic hatchback that I could get for a few hundred dollars and then rip into it with the angle grinder. If you are going to replace the structural integrity of the body with a tube frame, then you can quite happily cut off almost every part of a monocoque car except the floorpan, firewall and strut towers. In fact, you may even find that you don't need to bother with the tube frame if you get a bit creative with the angle grinder. Throw away the doors and maybe cut some hatches in the roof to pop up from or whatever it is that you need. If you wanted 4wd, then you could even start with an old Subaru of some description to give you extra off-road capability. Or if you could find something going cheap you could even start with a proper 4wd. I've seen old 4wd utes being sold for about a thousand bucks or less, which would do everything you need and more. You probably wouldn't even need to do much to the body, just paint it in NATO green and away you go!
There is some kind of workshop in Adelaide who seem to specialise in building buggies from cut-down cars and then trying to sell them for a fortune on eBay. Here's their website:
http://www.risingsuncarservice.com.au/Buggies.htm
I would recommend copying their techniques and doing it yourself rather than paying their rip-off prices. Enjoy!
Notice the way that they have kept the floorpan, firewall, strut towers and inner wheelarches. The inner wheelarches support the strut towers and then a roll cage tube extends to brace the top of each strut tower and tie it into the roll cage. They've also kept the door sills, as these are an important structural element of the car. Here's another example:
http://www.frontrange4x4.com/forums/showthread.php?11001-Subaru-Buggy