You can certainly choose your own auto-repair shop. You have the right to have any shop repair your vehicle for you after a car accident. The difficulty will be that if your shop says something was damaged that conflicts with the insurance adjuster. Initially, the adjuster will provide an estimate on the damage to your car. If your body shop thinks the damage is greater, they need to call the insurance company adjuster and talk to them about the additional damage prior to starting work on the vehicle. You do not want to be stuck between the body shop and the adjuster.
I recommend to my clients that the auto repair shop call the insurance company adjuster and get approval for the additional costs. This is normally called a supplemental report or supplemental estimate. This is normally not a big deal. The difficulty can sometimes be when your shop wants to use new parts from the manufacturer and the adjuster wants you to use cheaper off brand parts or used parts. The main argument for used parts is that the parts were not new in your vehicle so you are not entitled to new ones now. This however, is not much comfort to most consumers since they do not know the history of the car the parts were in.