I saw Light in the Piazza and was surprised that a mainstream musical had full songs in a non English language (especially one that wasn’t Spanish). I speak some Italian so I was pleasantly surprised. I know the 2009 West Side Story let the Sharks speak Spanish. Are there other examples of this?
While I've never seen 'Thoroughly Modern Millie', I believe it has scenes (or at least dialogue?) and a song or two in Chinese. The show's use of Chinese (both Cantonese and Mandarin) has become more controversial in recent years though.
I'm surprised nobody has menitoned the song "Sh'ma" from Parade, although it's a short song/scene, it's still fully in Hebrew I believe. I don't know if we can count "Fiddler on the Roof in Yiddish" under this, but it is fully in another language.