There is no way to ensure a camp. They are first come/ first served. Hell, it's not unheard of for a group to send down a member of their group several days early to hold down one of the Onion Creek camps for the weekend. Of course, I do not condone such greedy behavior.
That being said, I don't think that the White's camp is quite as sought after as the Onion Creek camps. Maybe you'll be able to get it.