There are several campgrounds right outside the main entrance. We were lucky and got into Trailer Village, and it is the best place, as you can get to everywhere via bus, and it is really close if you want to drive around. Outside the entrance is the next best choice.
If you can be fully self-contained, there is another nice campground right next to Trailer Village, but no hookups at all.
Before giving up completely, walk-ins at Trailer village are a possibility, but you have to be there and ask. We called the campground direct, and were able to get in one day later than we had planned, but were able to stay a week!