What is the name (and scene) of a gangster movie (think Goodfellas,
Godfather, etc) in which someone is driving a car and then asks
someone else to take the wheel as he leans out the window to fire his
tommy-gun while the car continues to move?

I've seen several movies using this action technique. One the the
sci-fi flick "The Hidden", and I forgot if that happened in "The Big
Hit". But since you mentioned a tommy gun and gangsters in Al Capone's
era, could it be something like Dick Tracy? Just a guess.

Was the one you're thinking of "O Brother Where Art Thou?" when
"Baby-Faced Nelson" takes the Soggy Bottom Boys in his car and has
George Clooney's character drive while said gangster shoots the
coppers behind him?

Isn't there a scene like this in "Bonnie and Clyde"?