Even though it is extremely frustrating to you as the accident was not your fault, the other driver’s insurance company will do a liability investigation to make sure their driver was at fault and that they owe you for the claim. This investigation will almost certainly include getting a statement from their insured. They may also want to speak with any witnesses and get a look at the police report if one was made. This all takes time, so rarely will another driver’s insurance company agree to provide you with a rental car in the first 24 hours after an accident.
If you have rental car reimbursement coverage on your own automobile insurance, you should consider getting a rental car under your own policy rather than waiting for the other driver’s insurance company to complete their investigation. If they determine their insured was at fault, they will reimburse your insurance company.