The vans are allowed anywhere on island (except Polihale State Park and other dirt roads, unless specified). There are six county campgrounds, one of which allows car camping. There are also three private campgrounds (in Haena, Anahola, and Waimea) and one state campground which all allow car camping. Private campgrounds typically have hot water showers and other features.
For more info, please visit The Kauai County Website.
Also, please note that you can go to any camping site with the camper as long as you set up the tent outside the vehicle (we provide a tent free of charge).