I actually started to look into this, kind of as a joke. A friend of mine in Washington state is rallycrossing her WRX. She won an event, and I asked if there was a FWD class. There are 3 FWD classes.
I am sure you could go out and have a blast on a totally stock car, so long as you didn't mind it getting dirty and a little scratched up. Rally tires would be the biggest upgrade you really "need" to make. That, and a good helmet.