Minor car accident (not at fault) - should I go to my insurance or small claims court?

Hon, if you take it to small claims court they will want to see the accident report, so no, you can't do that unless you file a report. If you want your car taken care of then you should make a police report of the incident and report that you tried to resolve this but they are avoiding you. Talk to the police and tell them what happened and be sure to tell them that the people refused to provide insurance information. In most instances that may be against the law. And then contact your insurance company.