If your vehicle is not drivable, odds are that it was towed from the scene of the accident to a tow lot. In Missouri and Illinois the tow lot, by statute, can charge storage fees for each day that your vehicle is held in their lot. The good news is that the at fault insurance company will pay for these storage fees for you. If you have to make a property damage claim through your own insurance, they will also take care of these fees.
You do, however, have a responsibility to make sure that the insurance company is not left with a substantial bill by causing the insurance company an undue delay in being able to get to the vehicle. This is called a duty to mitigate damages. It simply means that you must take reasonable steps to minimize the amount of damages that the car accrues by notifying the insurance company within a reasonable time that your car is at a tow lot.
You can fulfil your duty to mitigate simply by letting the insurance company know where your vehicle is immediately after the accident. Do not wait 15 days and then decide to call them if you already know the at fault insurance company's name. Sometimes you will not know the at fault insurance company's information until after the police report comes out. This can take 7 to 15 days. You will be ok in this situation so long as you notify them within a reasonable period after discovering their identity.
I represent clients for their bodily injuries only. I do not represent them for their property damage for several reasons. First, it would be difficult to fix or replace your vehicle if you had to pay attorney fees with what was recovered. Second, as the legal owner of the car, the client is in the one who has the legal authority to act for it. Finally, if the client has possession of the car, they are also best suited to handle the logistical aspects of getting the car fixed.
At the beginning of a personal injury case, I will send correspondence to the at fault insurance company letting them know that they can contact my client SOLELY to discuss property damage resolution. Adjusters are not to talk about the cause of the accident or my client's injuries. Most adjusters respect this without issue.
A little-known fact is that towing companies can keep your vehicle if no one comes to pick up within a reasonable period. Before this can occur, the towing company must provide the last known owner written notice that they are in possession of abandoned property. After the 30-day window passes, they then can apply to the Missouri Department of Motor Vehicles to have the title transferred to them as abandoned property. If the abandoned owner fails to come forward, they will be able to transfer title into their name and sell the car.