You should report the person that hit your vehicle to the authorities for not having insurance on their vehicle, which likely your state law requires. You can do this by involving the law enforcement agency that took the police report or a state agency, such as the DMV typically. State laws will differ about who to contact and what steps to next take.

In Florida you need a crash report and then you can request insurance information on the other party by contacting the Bureau of Financial Responsibility. Your state might have a like agency to contact. I would do so just to make sure the person does not have auto insurance coverage with a new company of which they did not inform you about.

If you find out that the person is uninsured, thus does not have insurance of any kind, and you have either collision coverage or uninsured motorist property damage you can put in a claim with your own insurance company. If you do not want to use your insurance policy or do not have these coverages, many states allow you to take the at-fault party to court to try and obtain a judgment requiring them to reimburse for your damages and expenses.