Can you be a vet without euthanizing any animals?

Yes, but there are very good odds you will not be in private practice if you want to avoid euthanasia. Potential careers options include clinical or anatomic pathology, epidemiology, public health and regulatory medicine.

However, veterinarians in private practice will at some point be faced with the necessity of euthanizing an animal for humane reasons. Allowing an animal to die a "natural" death often means leaving that animal in severe pain while it slowing dies of starvation or dehydration.