Be persistent not just with your vet but all the vets in the area. I usually send out 5-10 emails and hope for a response. Understand that vets are busy and they might look at your email with the intention of responding but then forget. It's important to call or email if you don't get a response say within a week and half. If you get no response after 2-3 of this, I'd say move on and ask the veterinarian.
In response to your particular situation, I might ask the veterinarian, "Do you have any volunteer or shadowing opportunities available?" In this way, it is a personal question about yourself without making it about you. This way if he or she says yes or no, it would not be as if you are using your relationship as a client to get this position.