The reason this works is that nothing has to be learned. Baa Baa Black Sheep is already in their head, so the only new thing is holding the puppet up and singing it out loud.
My shy one would not perform anything for months. She sang an entire rhyme from behind a cardboard box within ten minutes, because the box was doing the being-looked-at, not her.
We now have a bedtime version. Two rhymes, two puppets, lights low. It has replaced about half of the stalling that used to happen at bedtime.
Tip
Use rhymes they already know cold. A brand new song turns this into a lesson, and the whole point is that the words come out without effort. Sit down when you become the audience - standing over them makes it feel like a test.