The maple springs camp ground north of the river is the lesser used more rugged camp ground. It is still a car camping type affair though. And I think there is a feed
There is the other campground right there by the central hub of the park. Also a fee.
I have never camped at either one however.
Maybe if you are feeling froggy you can hike to one of the closer back country campsites; Three Springs, First Creek (there are two), or Homestead. No fee to do this but you do have to get a permit from the visitor's center and let them know which site you will be camping at.
I can't tell you much about RRG as it has been a long time since I have been over that way. And if I remember correctly we only hiked to Natural Bridge.