"I'm taking my dog to the vet."
Literally, of course, that means I am taking the dog to a person, not to a building or office or practice.
We might say "the vet's office called to let us know the heartworm test was negative."
In the United States, there are "veterinary clinics" and "animal hospitals." There is actually a distinction between them, but few people know what it is and use the two interchangeably.