There are plenty of car rentals and also off-road-car-rentals in Iceland.
Before you set off in Iceland, make sure you have the right to take the roads you intend to drive on. Many rental companies apply various restrictions: in the water often insurance does not work and the customer is charged with the cost of repairing and towing the vehicle, you cannot use the help of other vehicles in case of trouble and you have to wait for (expensive) help called by the rental company.
If you decide to rent a vehicle - buy the full insurance and support option in case of trouble. - You will definitely get a much better and better equipped car.