If you were in an accident and the owner of the other vehicle will not give you his insurance information is there a way for you to find that out?

yes usually several ways depending on the state.

1 if you have a police report that will usually show the parties insurance co

2 if you have a drivers license # or a plate # you can usually file a request with the state dmv to find out

3 if you dont have luck with #1 or 2 , sue the driver and the vehicle owner(if not the same person) , when they come to court the judge will compell them to provide proof of coverage or pay out of their own pocket, that almost always results in them coughing up the info as they dont want to pay out of pocket .