There are a few clinics in the town I’m doing my undergrad in, but I’m not sure how to approach them and ask for a shadowing opportunity.
I'ma not speak for other vets, but if someone wanted to shadow me the criteria they would HAVE to evidence are:
-- professionalism: dress decently, don't be pulling your cell phone out every 5 minutes, etc.
-- courtesy and respect for me and my staff: the truth is - we matter more at work than you do. remember that. we're here for us and our clients; any benefit we have to offer you is way, way, way down on the priority list.
-- boldness: you want to shadow? great - just tell my front desk staff and ask if you can meet with me.
-- not too much boldness: don't come asking to shadow me 3 times a week for a year. holy hell I won't be able to say no fast enough. ask if you can shadow for a day. if it goes well and you like me and you don't royally annoy me, then ask if you can do it again. after a few times, ask if you can do it regularly (because if I haven't kicked you out by then you're probably ok). remember that the first few times are dates; don't try and marry me right away.
-- personality: have one. be funny. be interested. ask questions when we don't look like we're going crazy. don't get offended when we say "not now" or "stand back" or "no, you can't go in THIS room with THIS client."
(We don't take pre-vet shadows where I work because of how chaotic and potentially hazardous it is; I regularly have 4th years doing externships, but that's about it.)