You have asked a good question. Generally, the reason zoos and nature centers don't hire outside entertainers is because they have their own entertainment. The zookeepers will choose some live animals and put on shows quite often for school groups.
However, if you live close to a zoo or nature center you could possibly promote the idea of a field trip bonus and provide a nice photo promoting your show. They could then offer it to school groups that are coming as a bonus for X amount of dollars. I've never tried that.
Another option is to figure out their busiest season and offer to come 3 times that season for the price of two shows. (Summer is probably their busiest time in the northern hemisphere.) Better yet, offer to come every Friday for two months at a certain time.
A better venue to try would be an amusement park, such as Six Flags. They generally have a small stage and would like entertainment during the summer. Have a video ready.