I have stayed at Mauna Kea (I was actually there for a camp with kids, so we had a few more resources than you'll have just camping.) The cabins were reasonably nice and I think there are actually showers in them. A couple of things: first, it is off Saddle Road and some of the rental car companies won't let you drive there. Second, it was quite cool at night, even in the summer. Pack a sweatshirt. It's pretty remote, but some nice scenery and we did some hiking up there.
I've also stayed in some little bird-house looking cabins at Volcano -- Namakani Paio -- you rent through Volcano House Hotel. They were small but comfortable for 4 .... I think there was a bath house outside. Again, it was chilly ... you'll want more than a blanket.
I think I camped at the old airport park once -- but it's been a long time ago -- and it might have been somewhere else in that vicinity. If you are going to camp, turn the window of your tent toward the ocean so you can get the cross breezes. I still remember a night camping in Hawaii as the hottest I've ever been in my life!!