I'm a Sound Design student, would I be allowed to bring my field recorder/boom mic and record animal sounds for a school project?

To be on the safe side call or visit the zoo first let them know what's going on They would be happy to talk to you

Agreed. Best to call the zoo to be sensitive to the animals

Interesting question. Safe to assume a boom pole into the exhibit is a big no no without permission and staff with you. Bringing a hand held directional mic should be fine though.