There are several ways you could do this, but I would think one of the best long-term would be to put the rubber chicken so that it is sticking out of the drawer in a small dresser, and then automate closing, and opening that drawer. So that it closes on the chicken. You could probably even set it up so that there were different closing positions, which could then make different sounds.
I’m not great at this kind of maker project, I just know that it’s possible. People in the past have definitely automated things like projector doors closing and cabinets closing. So I’ll leave that discussion for people who are better at that skillset.
I would probably avoid using anything with magnets to make sure that the donkey couldn’t accidentally get to it and swallow it. But you should be able to use a regular actuator.
@johnconstantelo or @blkwll might have some ideas.
And here’s a thread that discusses the idea as it applies to several different kinds of use cases