The NPS runs both Desert View and Mather Campgrounds, the only above-the-rim campgrounds on the South Rim, as well as Ten-X Campground, a few miles south of the South Gate Entrance. Year-round reservations for Mather and Ten-X can be made online up to six months in advance, but the recommended Desert View (mid-April to mid-Oct) is first come, first served.
Campgrounds, campsites and at-large camping below the rim are also run by the NPS. You cannot. however, book these online; to camp below the rim, you must secure a backcountry permit from the Backcountry Information Center.