On the South Rim, campground reservations can be made for NPS Mather Campground, located within busy Grand Canyon Village. (historic district, lodges, shuttle buses, visitor center, train etc.) There are no RV hook-ups at Mather Campground.
Mather Campground reservations can be made through the National Recreation Reservation Service. Call 1-877-444-6777 Online: https://www.recreation.gov/ (Reservations can be made up to 6 months in advance)