No, if the agreement has already been signed by both parties, then apart from the below, they cannot require a co-signor; that is, once an agreement is agreed to and executed by the two parties, one party may not after the fact impose new conditions.The exception would be if the transaction is contingent upon financing and your son cannot qualify for financing without a co-signor. In that event, if he doesn't bring in a co-signor for  the loan, he will not get the money; if he doesn't get the money, then he can't keep the car--unless, that is, he finds the money elsewhere to complete the transaction (e.g. some relative loans or gifts him the money for the vehicle).