Like the previous poster said, Door County can get crowded.
I'm not too sure about the private campgrounds because we only camp at state parks. The most popular state park there is Peninsula. We are not particularly fond of going there in the summer; not so much due to the crowds but because the swimming is not great at Peninsula. However, it is a beautiful park and has some great biking and fun activities. If you use ReserveAmerica, you should be able to find a site.
Also, Door County has had some problems over the last couple of years with high ecoli levels at some of their public beaches on the Green Bay side. That being said, we love going to Door County in the early fall. Still crowded but we do not have to worry about the swimming issue.