Can you bypass the electrical actuator motor for the heater on a 99 Taurus?

I'm quite confident something could be invented/improvised to take the place of the actuator motor, but...

By the time you put that much effort into it, and that much time scrunched up underneath the dashboard, you really could much more EASILY replace the broken motor.

Check with you local car salvage / junk yards and I expect you'll find a ready supply of used ones for $5. Even a new one is $30 or so.

See "Related Questions" below for more on how to replace the actuator - it should give you an idea on where it is and jump-start the process of bypassing it, or fixing the broken one.