Summary of article from The HIPAA Journal, by Steve Adler:
HIPAA does not apply to veterinarians because they do not conduct electronic healthcare transactions for which the Department of Health and Human Services has adopted standards, thus not qualifying as HIPAA covered entities. However, veterinarians are subject to various state-level data privacy and breach notification laws that resemble HIPAA regulations. For instance, California law prohibits the unauthorized disclosure of information concerning animal patients and their owners, with specific exceptions. Additionally, veterinarians handling data of EU citizens must comply with the GDPR. The American Veterinary Medical Association (AVMA) provides guidelines to help veterinarians navigate these diverse data privacy regulations.