Like someone said, smaller zoos will consult with local small animal or exotic vets. Where I live, the larger zoo has an on staff veterinarian (maybe more than one?), however they also consult a local vet who is board certified in Avian medicine and who runs an exotic/small animal clinic. I beleive he goes up there whenever they have a problem with their birds, but its not a regular thing. This same clinic also consults for a small zoo specializing in birds. They don't have the funds or need for a full time vet, so they have a vet tech on staff and one of the two vets at this clinic visits twice a week, consults, does examinations, and runs/ships off all of their labs. Also, if an animal needs to be observed or needs some emergency treatment, they go and stay at his clinic. Finally, since they are the only ones specializing in birds and one of the best with exotics, they also get a number of animals brought up from the wildlife rehab center about an hour away. They've had a number of interesting things from eagles to beavers.
So it's definitely possible! If your heart is truly set on it, I would recommend getting board certified in either exotics or avian. Zoo residency is a little more challenging and competitive, and if you get it you might as well try for zoo med.