Hiya Sadly, no, you need to use 'veterinary surgeon' instead. 'Vet' is a bit informal for a personal statement. Which is a shame because it uses up so many less characters!
I thought that if you use Veterinary Surgeon or Veterinarian once at the begining that it would be okay to abbriviate it to Vet for the rest of the statement. This is what I've done and I got it check by a Vet Student and she said that it was okay to do so.
you'll sound repetitive if you keep using veterinary surgeon! mix it up a bit but keep it formal. at the end of the day i don't think it'll be a deciding factor as to whether or not you get an interview
Obviously it would be ideal to have everything written out in full but if you're really struggling for characters, evidence would suggest that you'd be fine to use the word "vet" a few times!
I asked a tutor in my college and she has close links with liverpool uni in the vet med and medicine bit and she said its alright to use vet seen as most vets will refer as themselves as a vet.