The Producers which opened in 2001 had its movie made in 2005 so that was a little over 4 1/2 years later.
Others that were close:
If you count the recent revival of
Chicago, which opened in November of 1996, it was 6 years until the film.
Not a musical but a play,
Proof (2000), had a movie 5 years after it opened.
The movie of
Little Shop of Horrors opened in 1986, which was 4 years after the show began off-broadway in 1982.
"You pile up enough tomorrows, and you'll find you are left with nothing but a lot of empty yesterdays. I don't know about you, but I'd like to make today worth remembering." --Harold Hill from The Music Man