The insurance carrier of the individual who struck your vehicle is under no contractual obligation to provide you with a rental car. You do have the right to sue the other diver for damages including personal injuries, if applicable. If you have no collision coverage of your own, your claim would also include demand for compensatory damages for your damaged vehicle and for your out-of-pocket expenses arising from the loss of use of your vehicle.
The other person is responsible for loss of use of your vehicle for a reasonable time. Therefore, the insurer is responsible for the rental car. If they will not pay, you could file a small claims action against the driver. The insurance company would then have to reimburse you. These are general rules and I would check with a lawyer in your state to verify the accuracy in your jurisdiction.