For those unfamiliar with the term "diegetic music", it refers to music that's actually played in the story rather than just for the benefit of the audience. The characters can hear it too.
With that in mind, are there any musicals where the characters acknowledge that they're constantly breaking into song? Buffy the Vampire Slayer's "Once More with Feeling" is the best example I can think of for this.
A lot of stage and movie/TV musicals where the characters are actually singers have prominent diegetic elements - so Beautiful the Carol King musical, Jersey Boys, Elvis a Musical Revolution, The Commitments, Fame etc.
In the recent Star Trek: Strange New Worlds episode "Subspace Rhapsody" all the characters were aware that they were breaking into song, much like they were in "Once More With Feeling". Which isn't surprising as OMWF was a very clear influence on the Star Trek musical.
Although I love musicals, I have not seen a wide enough variety to know of a fully diegetic musical, only musicals where some of the songs are diegetic, with the possible exception of Urinetown where the narrator characters imply everyone else in the show knows it's a musical and therefore are aware of the songs being sung.
LITTLE SALLY: But aren't you scared the rebels will see you?
LOCKSTOCK: Oh, I may be a cop, but I'm also the narrator. So no one can touch me, not if they want the show to end.
So I went and did a brief search and found some answers on TVTropes.org! Some of the responses out of the Theatre section are worth looking at, especially since one of the entries (Camp Rock, under "Films — Live Action") is also a full musical you can license from MTI and put on.