Hopefully your questions/concerns will be answered tomorrow in the live stream,since it will be all about the vet clinic :)
There was some leaked footage that came out around the time that the Create A Pet footage was released that showed the vet career...from what I remember (I only got to watch it once before it was taken down) the owners of the pets seemed to stay in the waiting room area but I think the pets wondered around a bit? But that also wasn't the final version of the game either.
If it ends up being an issue where pets/Sims are wandering around the clinic,I guess technically you could work around it by locking the door for employees/household and then unlocking it when you have to go get another patient. But we won't really know if it will be an issue until we see some the live stream/get the pack ourselves.