On my 1990 535, the defroster and mid-level vents are actuated with a cable, but the foot vents are electrically switched. Since you have hot air flowing through the car, I would guess it is a flap somewhere that is not opening. Unfortunately I don't have a magic bullet in mind besides good old investigative surgery armed with your multimeter.
The electrical schematic in Bentley is a good starting point, as is the factory service manual which you can probably get on Ebay. Comes on a CD and it's BMW's official manual. There is also an electrical troubleshooting manual that has all the info you'll ever need. Other than that, maybe someone else can post with a more directed answer...
Robin
72 Chevy K10
01 E39 M5