If I'm understanding the situation correctly, my concern is that it really sounds you are being asked to complete your training off the books ("volunteering"), then potentially be hired once your training is finished. In my state, that would be illegal. There is an option to pay employees ages 16-19 half of the legal minimum wage during the first 90 days of employement (training), and minors 16-17 can be paid 85% of the legal minimum wage regardless of training status until they hit 18. Other than that, any time you are considered training, you are to be paid accordingly. If you are not under 19, you can't be paid less than minimum wage during training.
I'm trying not to generalize across the field here, but there are clinics out there that abuse the system/pre-vets who need experience and end up getting a lot of free labor out of it. That may not be this clinic's intentions, but it's something I always looked out for. Hell, one of my working interviews was about 6 hours long, and I did a lot of heavy cleaning. Never heard back, but I did hear from friends who applied for the same job that they all did the same thing. This also isn't a problem that is unique to vet med.
There's a difference between someone wanting you to volunteer so he/she can get to know you better before agreeing to hire you (which I still don't entirely agree with) and telling you to volunteer so you can learn the ropes before you are paid. The wording of the latter would probably be illegal. In general, unpaid work in vet clinics can be tricky legally and makes use of loopholes sometimes. Also, interns and volunteers aren't the same thing.
TL;DR-just be careful. If it feels wrong, it probably is.