I saw lots of car campers - and I was lucky enough to meet The Novel Traveler while I was there. I didn't see any tents but I don't see any reason you couldn't set one up in good weather. I left when the wind was blowing so hard I thought it might rip the door off my trailer. Could be a fluke but it was some serious wind. Other days there was no wind.
Yes, I did for two weeks. You can also set a tent out if you like. Some people do it at or around the picnic benches. There are bathrooms, cold beach showers. Indoors and outdoors.
It can get pretty windy some days. So. Check the forecast before tenting. Indianola might be better for tenting. 2 miles down the road and has grass.
I enjoyed my stay. Met cool nomads there.