Can your insurance company do anything if none of the drivers involved a car wreck gave information?

It really depends on how reasonable they are. If you are in an accident and the other party or parties refuse to give information regardless of who is at fault, then CALL THE POLICE because a police report is the best weapon against anyone involved in the crash trying to change their story later. If you're not at fault for an accident and you have no other information on anyone else, your insurance company may be suspicious about that and could quite possibly reject your claim or pay it but make it a chargeable account to YOUR POLICY.