I rented a Jeep Cherokee and rented a sleeping pad and sleeping bag from a camping supply equipment place in Reykjavik. We did the entire Ring road plus westfjords in 12 days. I recommend an SUV over a camper van if you're watching your budget. Petrol is very expensive in Iceland. It was over $100USD per fill up. We also rented other gear so we could cook and stuff. The SUV was really comfortable and I was plenty warm during the night.
The camping facilities are good and abundant. Some are open all year and some are only open during certain times of the year. The have spots to park your car, hot water, and some have inside hang out spots with laundry. You might have to pay for hot water, though, so just go to a local swimming pool to shower off. You'll pay more at the pool, but then you get to soak in the hot tub before you head out on your journey for the day.