Fortunately for us, Glenn Miller made two jukebox films before he went into the Army in 1942—Sun Valley Serenade (1941) and Orchestra Wives (1942). Fortunately because his band was featured in both.
While they weren't playing live to camera (songs in films typically were recorded earlier and played through monitor speakers while the band played along on film). This avoided the risk of errors and retakes while filming and allowed actors' "vocals" and "trumpet solos" to be overdubbed by pros.
For example, on the female vocal side in Sun Valley Serenade, the voice you hear belongs to Pat Friday, who overdubbed for actress Lynn Bari's vocals. Here are videos from both films:
Here's Moonlight Serenade, the band's theme, and I Know Why and So Do You from Sun Valley Serenade (colorized). Bill Kirchner sent this clip along yesterday, which shows the reed configuration for the Glenn Miller sound...
Here's In the Mood from the same film...
Here's It Happened in Sun Valley from the same film...
Here's At Last from Orchestra Wives...
Here's Serenade in Blue from the same movie...
And here's I've Got a Gal in Kalamazoo from the same film...