Is there a union or association to which vet employees must or can belong to?

To the best of my knowledge, there is not a union specific for veterinary clinic workers. Because the veterinarian is considered a manager or supervisor within the clinic, the vet would not be eligible to join or start a union. There may be clinics were the vet assistants, vet techs and/or other support staff are part of a union, but this would be unusual.