In most states you should be able to contact the state's insurance regulatory agency with the information you have plus the police report on the hit and run accident to try and receive the insurance information for the at-fault party.
For example in Florida you contact the Bureau of Financial Responsibility with the crash report and they should research and reply to you within 10 days. In California you can request the insurance information by contacting the DMV's Financial Responsibiity Unit and submitting a form.
In New York you fill out form FS-25 which is a Request for Insurance Information for NY Registrants Involved in an Accident and is available on the NYS DMV web site. You then send in the form with a $10 fee and the DMV will try to verify the insurance coverage of the vehicle.