I wouldn't be surprised if Actors Equity and the Broadway unions didn't have a prohibition against using a 100% prerecorded vocal track. While I know that some shows use a certain amount of prerecorded tracks, it's one thing to be playing the prerecorded voice of the actress playing Christine Daae while she and the the Phantom scramble up or down a ladder backstage in order to effect a scene transition, it's another thing completely to reduce the Broadway experience to something akin to a theme park stage show. The Broadway musicians union kicked up a fuss about Priscilla, Queen if the Desert using a prerecorded strings section (I believe strings were used in just one of the songs in the entire show) so it's hard to believe that the performers unions would go along with a fully recorded vocal track.
Even if you ignore the fact that securing the "normal" lead performers schedules would be both expensive and challenging, and that Disney's practice in creating a Broadway show would surely include planning on being able to take the show around the world - including the option of staging the show in a number of countries simultaneously, many characters are taxing on a performer's vocal chords (Animal, for example). This, combined with the fact that most of the Muppet performers play more than one character (often more than three!), and that Broadway shows occur eight times a week, all suggest a logistical nightmare far beyond that seen in the character-sharing of Avenue Q that would be unsustainable by the regular performers. So if, by some miracle, the poor box-office performance of Muppets Most Wanted hasn't put an end to the prospects of Muppets on Broadway, I think you could guarantee the roles would be performed by non-regular Muppets actors.