The AVMA has provided a summary of state rules related to whether a veterinarian must write a prescription at the request of a client, in lieu of dispensing medication.
Per the Principles of Veterinary Medical Ethics of the AVMA, veterinarians should honor a client's request for a prescription.
The AVMA State Legislative and Regulatory Affairs Department compiled a list of states that address the matter. Of the states on the list, 19 have laws requiring veterinarians to provide clients with written prescriptions on request.
In addition, the Massachusetts and North Carolina veterinary boards have policy statements requiring veterinarians to provide written prescriptions on request. South Carolina regulations indicate only that a veterinarian may provide a written prescription.
Vermont is the only state to specify that it does not require veterinarians to provide written prescriptions on client request.
The "Written prescriptions" summary report is available here by looking under the heading of "Pharmacy and vaccination" and the subheading of "Prescriptions."