Who your letters come from is actually based on what school you are applying to. Some schools want 2 from veterinarians and one from somebody else, some schools want 1 from a veterinarian, 1 from a professor, and 1 from somebody else, etc. Ohio State is an example of a school that wants 2 from veterinarians and 1 from anybody else. Ohio State also explicity specifies that you do not have to have worked under a veterinarian who is writing you an LOR. Of course, they will be able to write you a stronger LOR if you have worked with them and they know more about you, but that's up to you. The vets that are writing my LORs haven't known me too terribly long-one for a year and one for a few months-but I've worked with them extensively this summer so they both have a pretty good idea of who I am and what kind of candidate I am. The other person I chose was the stable manager at a horse farm where I worked for about a year. She really liked me and I know will write me a very positive letter.
So choose your references wisely and based upon the school's requirements. Instead of asking if they will write you a letter of recommendation, ask it in a way that 1.) specifies that you want a GOOD letter of recmommendation, because vet schools have gotten letters saying "oh dear Lord don't let this person into vet school" and 2.) lets them have an "out" so you don't put them on the spot of having to tell you they can't honestly write you a positive letter. A good way to ask is to say "I was wondering if you feel like you know me well enough to write me a positive letter of recommendation for vet school." That way you've specified that you want it to be positive and you've given them an out if they don't want to-they can just say "You know, I don't think I know you well enough..." blah blah blah.
Good luck!