The original mission is indeed for Allen, Arya, Oz, and Tyki. They're going to Disney World! Specifically, they're going to MGM Studios in 1993, for the New York Street show to pose as the Teenage Mutant Ninja Turtles. The Turtles originally open the show, with a short 'ninja dance', after which they exit the stage and disappear until after the show. This show happens to run approximately precisely during the time there are fewer people working around the Muppet Show's performances, leaving ample opportunity for the four to procure one of the back-up muppets of Miss Piggy; the performance ended long enough ago for there to be fewer stagehands and muppeteers nearby, and the next isn't for long enough that anyone's returned from lunch. All in all, pretty straightforward.

The Initiative however, while being relatively confused that the four of them can't return simply with a muppet, isn't averse to allowing Break to go with Lenalee and assist them when it comes to them apparently requiring back up.

The Initiative does monitor the psychological status of its recruits, and the lack of any amount of monitorable teamwork amongst any of the six of them who end up going has been noted. However, while this is an attempt to remedy the situation, it also serves another purpose. The emergency replacement procured by Disney will later fall apart during a show three years from then, disillusioning several young children and in particular, one that would have grown up to later work in stagehandling herself. Instead, she winds up going into animatronics, producing a specific political satire that damages a politician's running in polls just enough for him to lose a local election.

You guys. Come on.