Your radio will do just fine. When you get there, they'll just tell you which station to tune into and that's about it! They have a short range antenna so basically they only broadcast the movie sound far enough so you can pick it up on your car radio while in the drive-in theater.
Also some people have trouble with their car lights because some cars have parking lights. You might turn off your car, turn on your radio and see that your headlights are still on. Just pull up the parking break and the your lights will go out.
Personally I DO bring a battery radio just because I go so often that I have killed my car battery. It is fine if you go once in a while so don't worry. Although at my drive-in there are usually two movies playing and the drive-in employees always encourage the audience to start up their car and let it run for a bit in-between films so you're not in a situation where you have to get a boost to start your car.
