Can an autistic person become a vet?

Yes, an Autistic person can absolutely become a vet.

The photo that's currently showing accompanying this question is of famous Autistic person Temple Grandin, who despite her moderate symptoms went on to become a professor of animal sciences, as well as being a writer, activist, and an expert in animal behavior. So clearly people who are Autistic can go into further education and enter into animal care as she did. Obviously everyone is different so it depends on the individual's abilities, but veterinarian can be a good job for people who are Autistic as it means minimal human interaction and animals can be therapeutic. Although Autistic people have higher empathy so may struggle with animal suffering and if animals die or have to be put to sleep.