I have to respectfully disagree here, mostly because not everyone has 3-6 people that want to do whatever they can to get you into vet school. Some people might have 1-2 professors or vets that they formed close relationships with and would be willing to go the extra mile, but I would guess most people get your run of the mill "this person is good for x,y,z reasons" letter. If your LOR writers absolutely adored you and wrote a novel each about your merits, that's cool. None of the professors that I convinced to write me letters must have written anything too bad about me, because I got in. But that's just my wacky reasoning behind my opinion.
Bum advice.
Maybe the reason you got in was
in spite of your letters. I'm not saying it necessarily was, but it's bad advice to encourage people to settle for more generic recommendation letters unnecessarily, just because you had some and you got in. Nobody - especially not MB - is saying you
can't get in with less-than-amazing portions of an application.
You want to do the absolute best you can with every part of the application. Some parts will be better than others just because that's the way life works and most of us can't be amazing at everything. But I'm with MB: You want to have letter writers that <actually really> want you to get into vet school.
If you don't have them, ok, you go with what you have. Again, that's just life. But for people who are in the planning stages of vet school, it's better advice to suggest that they start building those relationships so that they have people who are more excited about them personally when it comes time to have someone write the letter.
In regards to how you approach someone, I've always strongly advocated in-person
if it's possible. It's more professional and it delivers the message to the person you're asking that you really care about the quality of what they write. So they are more likely to put more effort into it. If someone sent me an email with "Um, hey, could you write me a letter?" how much effort am I going to put into that compared to the person that shows up in my office wearing halfway-decent clothes with a copy of their resume and says "I'm applying to vet school. I felt like you would be well-positioned to write me a letter of recommendation because of our <whatever history you have>. Would you be willing to help me?"
Holy cow - I'm going to put out EFFORT for that second person.