It depends on what is being said.
If you said :
I am a vet
it would be
Je suis vétérinaire
but the passage says -
Un vétérinaire est un docteur pour les animaux... = A vet is an animal doctor ...
In fact, there is a sentence in the text that doesn't use the article -
pour devenir vétérinaire = to become a vet
I know it seems crazy but that is just the way it is...