Can an 11-year-old work in the zoo or volunteer and get paid?
You don't get paid to be a volunteer. As for working you would have to talk to HR at the ZOO and ask them about that some states you have to be over 14 to be employed. It could also be unsafe to work at a zoo in such a small age.
In most states, if not all, you have to be at least 14 ~ 16 for a company to be able to legally hire you, and if you're 14 to 15, you can typically only get a job in a VERY basic area (i.e. sign twirling). Specifically in zoos, though, the people who are hired to actually take care of the animals are required to have diplomas in zoology/etc. in order to be able to not only safely, but accurately, handle the animals. If you're talking about small vending stores in a zoo, though, you have a possibility of being hired, but most retail positions require you to be 16 or older.