Losses caused by damage to your car by another person who is at fault in the accident are covered by

The other persons insurance unless they do not have insurance if you have full coverage your insurance company will cover it usually including you pay your deductible and in both cases you can't take them to court by filing a small civil suit and making a report in the police station. Assuming you have a plate number or name that identifies the other party they will be obligated to go to court and if they don't they will recieve a bench warrant or also know as FAT warrant either way these things do take time but each step is a step closer to you winning your case and well deserved compensation