How can you become a vet without having to dissect animals?

This is an option, actually, although you will still have to demonstrate your knowledge and ability to identify anatomic structures on dissected specimens. You will need to work with the college of veterinary medicine you have been accepted at to arrange alternate teaching models in lieu of performing dissection or cadaver surgeries. I don't believe any veterinary schools have "kill surgeries" (where a live animal is put under anesthesia, students practice surgical techniques then the animal is euthanized without recovering from anesthesia) any more.

Having said this, you will have to perform necropsies on animals that passed away or were euthanized - this is part of being a veterinarian and is a valuable diagnostic service offered to production animal farmers.