Not "walk through" to say, but in the Lied Jungle in Omaha Indian fruit bats can (and will) whiz above your head.
I also believe Toledo, form what I've heard, has a walk-through flamingo exhibit, but I'm not sure if the flamingos can cross your path or anything
I know also SDWAP had a very interesting petting zoo up until 2011. I'm not sure what exactly was in there, but some images I found in the gallery and online show juvenile eland, wild asses and roan antelope along with adult deer (Not sure of the species) In 2011, according to their youtube channel, this cool petting zoo was replaced with the old stupid idea of goats![]()
Metro Richmond has a few blackbuck in their petting zoo area! Along with a fallow deer and the usual goats, donkey, zebu, etc