Does an insurance company have to pay for a rental car when repairs are being made to my vehicle?

The insurance company is generally required to pay for loss of use. In other words, you are not able to use your vehicle and they are required to compensate you for that loss. Typically, that is done by paying for a rental car, so yes you should ask the insurance company to provide and pay for a rental car so that you will have a vehicle wow your vehicle is being repaired.