Do you know that this vet takes shadows? One good way to find vets to shadow is to ask the pre-vet advisors at your school (or the nearest university - just call and ask, they won't know you're not a student there!) for the names of local vets who have been particularly amenable to students shadowing (and whom students have reported as supportive and willing to teach).
I had particular vet in mind at a practice that I knew took shadows. I wrote a letter directly to him... And waited a good couple weeks before giving up on ever hearing back. I tried not to be offended that I didn't at least get a "no", it's a really busy practice. So then, I went to the front desk at a time that wasn't too busy, told the head receptionist that I knew they'd taken shadows before and was interested, and asked whether they had some standard procedure for applying. She told me that one particular vet there supervised all the shadows, and I should write a letter outlining my experience and reason for wanting to shadow and she'd give it to him. I got a call back the next day.
(The whole time I've been there, vet #1 never let on that he had any idea I'd contacted him, but I certainly didn't mention it to him or anything either.)
My point is, although I agree with Angelo and pressmom that going directly to the source could me most productive in many cases, it might actually be the case that the clinic has an established program for shadowing, especially if as you say it's a large multi-location outfit. And it can *never* hurt to get on the good side of the reception staff by asking how things should be done. If you think they're giving you the run-around or just seem unfamiliar with the shadow concept, then you should try a letter directly to the vet before giving up on that practice.
Oh, and as far as who to pick... You'll want a letter of rec out of this, so it would be best to stick with one vet at any given practice, so they get enough experience with you to write a detailed letter. If you volunteer each week with a different vet, nobody will know you very well in the end.