You need to get a lot of experience, as much as you possibly can. Network, network, network.
The zoo community is very small and very close knit - they know everyone. It is very difficult to get into that community, and yes the number of job openings are VERY small.
You really need to see if the job description is exactly what you're looking for. If you can't shadow a zoo vet, then try connecting with them to see what their day-to-day is like, and see if that interests you.
I set myself on the path of zoo medicine: B.S. in zoology, worked at two zoos, networked, etc. I loved the keeper aspect of it but found out I wasn't too thrilled with the vet aspect of it. The vets I shadowed seemed to do more paperwork and necropsies than treating animals, not many surgeries, and lots of prescribing medications. Just didn't seem like what I envisioned myself doing for the rest of my life.
I'm also interested in the people/owner part of vet med. I like teaching people, explaining things, etc. so now I'm leaning more towards exotic animal/wildlife private practice with a little emergency medicine to keep me even more on my toes. I think that will give me the right balance of people/surgeries/handling weird and interesting animals.
I certainly wouldn't be adversed to being a part-time relief/on-call zoo vet, they need vacations too! I am still going to try to get a 4th year rotation at a zoo, as I am overall madly in love with my zoo experiences.
Key thing is, get your butt out there and get some experience!