I've been to a few places where I've had close encounters with animals but I'm not sure if that's what your question is after. I've fed an elephant, fed kangaroos and held a baby monkey but a lot of these places were wildlife sanctuaries so pretty hands on. I paid to swim with dolphins once which was an actual encounter type thing and that was beautiful. For some of the places I've travelled to, even though I've wanted to meet and greet certain animals my research showed me that certain 'sanctuaries' or 'zoos' were to be avoided due to the way they abused their animals. As a tourist I think it's important to give your custom and money to places that genuinely care for the animals welfare. There are a lot of attractions based around animals that are very cruel. I try to research places I visit on my travels so that I'm not just adding to the issue.
What's an animal encounter program? Do you mean through a school? My daughter's daycare had a program where they brought in fertilised eggs and hatched out chicks. The kids liked it. We've also had contact with native animals at Lone Pine Koala Sanctuary. The kids patted some kangaroos, like in the picture, which they enjoyed.
Years ago when my daughter was four and it was hysterical. She didn't quite grasp the concept that when an Emu demands your bag of food you should surrender it. She was yelling No at this demanding Emu and starts running with her bag of food. The Emu takes off after her and myself and a collection of interested adults took after them both. Worthy of a comedy sketch it did end well but the Emu was only sidetracked by a larger bag of food and my daughter got to keep her bag of food. As a parent there was the odd moment of terror but it all worked out well and is now one of those family tales which we all have a giggle over at family reunions.