No, the closest you'll get is doing the American Idol Experience in DHS. Don't know if they would give you a copy of your performance or anything if you actually made it to the Finale show or something?!
There is a booth at Disney Quest where kids can pick a song to sing along with and you get a CD of their "performance". It is one of the things you have to pay for separately and I don't remember what it cost, but DD did it on our last trip.
Years ago there were booths outside of one of the attractions at DHS where guests could try to voice some lines from a Disney movie. No way to save the experience but it was kind of fun.
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