I’m sorry to hear that you were involved in a hit and run, but I’m so glad that you were able to get the license plates! To figure out if the car was insured for free, file a report with the police and contact your car insurance provider—they can properly inform you on what will happen next in this process.
If you do hit a dead end, though, you can also try the DMV. You’ll need to provide your contact information, reasons for your request, and a copy of the incident report from the police. The DMV has the right to refuse your request, so it is really important to follow proper procedures here. But as long as you follow procedure, you should be able to get the driver’s insurance information.
Best of luck with everything!