I have personally never seen the state parks fill up. There was a time when we could camp just about anywhere along the road system in Kodiak. Much of that land is owned by native corporations. Due to irresponsible campers over the years, the native corporations started requiring day use & camping permits on their land. You can contact Discover Kodiak Visitor Information Center to get more information about the state parks and permit camping along the roads. http://www.kodiak.org
Budget and Avis are the two car rental companies in Kodiak. Car rentals are more costly in the summer time all over Alaska. You should check the prices out in advance. Something most people don’t consider is renting a U-Haul vehicle. They are typically cheaper to rent, but you also pay for the mileage you travel. There are only around 120 miles of paved and improved gravel roads in Kodiak.
Some extra information: The local laundromat has showers that you can pay to use. The parks don’t have those kinds of facilities.