Yes, a veterinarian can use video and audio recording during appointments. Statutory and case law allow such a practice.
However, to ensure that the veterinarian is in compliance with regulations relating to the practice of the profession, he or she should first obtain written consent from the client to perform the video or audio recording. Securing consent will protect the veterinarian from any legal claim (of exploration, for example).
It is important to note that the presumed intent in this case would be to use the video or audio recording for the purpose of diagnosing and treating the patient. If the veterinarian intends to use the recording for other purposes, then the written consent should reflect a clear explanation of those other purposes.