We have been there MANY times. There are two campgrounds on the island. The first is just as you cross the bridge at MM1. Great campground with water access. Not on the beach. The other is further down the island and only accepts motorhomes. Closer to the beach but still on it. We love Hilton Head and spend at least two weeks there every Sept. Since we are in a fifth wheel, we stay at the one just over the bridge. Both parks are very nice.
I'm headed there this February for a little wintertime RVing. It looks like you pick out your site when you get there.....they let you loose in a golf cart to check out the sites. I believe the individual campsites are privately owned. It looks like taking bikes is a good idea as there are bike trails all over the island.