If the owner is of an age when it's possible that the vet could sue them, the vet could if they wanted to in reality, but it's unlikely that they would. That's because being bitten is a risk which they will be fully aware of when they take the job, and they will also know that it's less a case of if they're bitten, far more a case of when they're bitten. Animals at a vet's may very well be much more likely to bite, such as if they're unfamiliar with the vet who they're seeing and are wary of strangers, or if the animal is in physical pain.
Vets are also taught how to restrain animals which they treat during their training, including using restraining tools such as a muzzle, plus they may also have someone else helping them, like a vet nurse for example.
Hope this helps.