They are both technically correct, but "Your car requires washing" would be the one most commonly used.

None of them sound natural. "require" sounds strange and too formal for a car washing. We would say: "Your car needs to be washed." "Your car needs a wash/washing." You should wash your car."

The most commonly heard expression is probably "Your car needs washing", which simply means "You should wash your car" or "Your car is dirty, so you'd better wash it". "requires" sounds like an unnecessarily strong word here as it usually means there is some person or organization that demands it.

washing