The racing commission would still be able to charge a fee for an individual's participation in racing in this state. No, though it would become smaller. The national license would be available only to those racing participants with uncontroversial backgrounds. Participants who do not meet the high standard for a national license may still meet the requirements in individual states. Ultimately, states would be dealing only with the problematical cases.
Some states have reciprocity agreements.
The difficulty arises when a state's Code requires certain checks or certain standards which a neighboring state's Code doesn't. In order to have true reciprocity, as with driver's licenses, all 43 states in which pari-mutuel wagering is legal would have to pass similar legislation. While reciprocity is convenient for participants, it doesn't always ensure each state that its own standards have been met.