If you want to go to veterinary school, getting work in a veterinary hospital prior to college is really a good idea. It looks good on your college application.
However, you would work as a veterinary assistant rather than a veterinary technician. Veterinary technicians (as defined by the American Veteriniary Medical Association and the laws of many states) are people who have a degree in veterinary technology and have taken both a state and national exam to prove their qualifications and have been credentialed as a "veterinary technician".
A veterinary assistant works for a veterinarian but does not have a college degree in veterinary technology and hasn't taken the state and national exam. Veterinary assistants are usually trained on the job though they may take vocational training courses as well.
The job description of both positions is the same in many ways with a few exceptions in tasks that can and cannot be performed and the level of supervision required for certain tasks.
Veterinary technicians and assistants take on many of the general nursing duties and day-to-day tasks to keep a veterinary clinic running. They restrain animals, monitor hospitalized and surgical cases, fill prescriptions, run lab tests, collect samples for testing, maintain equipment such as surgical instruments and anesthesia machines, interact with clients, clean cages, keep inventory, etc etc. They can fill any role that is necessary.
This is a great way for to get some experience in the veterinary medical field and ensure that it's where you really want to be.
I would think that there would not be enough business for a trainer to have one on staff full-time at just one vet practice, unless it's huge. I also find the vet's office to be a mentally stressful place for pets, and one that I would not want to take my dogs to for training. Personally, I probably wouldn't use a trainer located in a vet's office for that reason- I want training to be fun, low-stress, and being surrounded by sick and stressed animals does not accomplish that. My vet maintains lists of local trainers and the behavioral problems they are prepared to address and hands it out to their clients, I think that's effective and really the way to go.