It can also help if you already have experience handling large animals. I didn't have a problem finding several vets in the area to let me shadow, but I think that was because I'd been riding and showing since I was tiny.
So, it may help you to start out getting large animal experience by volunteering with a therapeutic riding program, working at a stable, etc to make yourself more marketable and show that you know how to put on a halter, restrain, jog, lunge, etc. My undergrad even had a horsemanship PE class. Do something that shows them that you're comfortable around large animals and have a basic grasp of how to handle them. Plus, any large animal experience can go on your application, so you're really benefiting in 2 ways! 😉
Most equine vets are insanely busy because they have to travel so far, and I don't know many that would have the patience to hand-hold you through basic horsemanship. Make yourself more marketable to them by going ahead and learning basic horsemanship, and I don't think you'll have a problem. (But remember to be honest if you only have limited experience, since you don't want to get your kind-hearted vet bitten or kicked!)