You do not need anything other than your camping reservations. The camp ground is located on Forest Service land. I see that you never asked about cars but just to be clear, you can have two. There's isn't room for more than that at a campsite.
I can't say I am correct, but I have only had the camping reservation personally. Reading the signs in the park give conflicting information. On 1 sign it mentions 2 vehicles are included, and 3+ are $7 per vehicle. But another sign indicates you must may pay per vehicle regardless. If you are real concerned you should call and ask.
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