Vet school is harder to get into than medical school, so anybody that's a vet really wanted to be one.
Animal people tend to be pretty weird anyway, and every vet knows that, so being an aspie will be less of a disadvantage than if you wanted to be a salesman or something.
If you can keep scared animals calm, and you can concentrate on doing your job in an environment that includes things like blood, vomit, injections, urine, feces, animals in pain, and the chance of getting hurt, that's going to count for a lot more than having a charming conventional personality.
If you're dealing with the public at all, you'll be talking to people about their animals, which you'll probably find a lot more agreeable than trying to make idle conversation.
What you said in your initial post is about all there is to it. You're interested in becoming a vet and you'd like to get some experience. They'll be noticing your aspie traits probably anyway. I wouldn't see a reason to bring it up unless you see a reason to bring it up.
_________________
They murdered boys in Mississippi. They shot Medgar in the back.
Did you say that wasn't proper? Did you march out on the track?
You were quiet, just like mice. And now you say that we're not nice.
Well thank you buddy for your advice...
-Malvina