You would need to go through a two year vet tech program and then take the exams to be licensed. From what research I've done, you need a B.S. to become a zookeeper and only an A.S. from an accredited vet tech program to be a vet tech at the zoo but having a bachelor's on top of it will give you a better chance. It is extremely difficult to get a job at a zoo vet clinic due to low turnover and limited positions available. You would have to get by on doing a bunch of unpaid (or small stipend) internships and volunteering at zoos as keepers before even consider applying. I recommend getting skilled at dogs, cats, horses, etc. first and then work your way up to zoo animals since most vets just base their diagnosis on domestic/common pets and then learn as they go.
Here are some zoo medicine websites for you to check out. Zoo medicine is also considered a specialty that takes 5 years to complete and then you take another exam.