Can They Search My Impounded Vehicle?

Unfortunately for you, yes.  They can perform a thorough search of your vehicle and all contents in your vehicle.  This applies even if your car was towed for parking in a fire lane or if your car was stolen and subsequently found by the police.  Unless there is a reason/cause (e.g. they find bricks of coke in your backseat), it probably won’t happen. The good news is that the police can’t just go around impounding vehicles for no reason.