How to repair the flap
Before you go tearing the HVAC apart, which by the way, will take every tool you own, involves drilling and cutting etc. The thing is, there are many parts, vacuum motors, a blend door and its actuator, and a really complicated electrical vacuum control. It's not just one thing. So, If you haven't already, try this....
It sounds like vacuum is not getting to the vent, heat and A/C controls. Find the vacuum can. It may be under the hood on one of the inner fenders. The "one-way" check valve is usually located in the line and close to the can, make sure it works properly. The valve could be the very thing that's causing the problem. Trace the lines back to the vacuum block at the intake manifold or fire-wall, and also back to where the line goes thru the fire-wall. Check the can for leaks or obvious holes. They used to be just a plain metal can, but it could be a plastic ball. Check the lines as you go, for breaks and rub-thru's. Move the controls thru all settings and listen for vent door movement. Worst case is a bad vacuum motor or air flow position selector switch, and/or a faulty Blend Door Actuator. If you have been driving around with the Temperature knob or control set at the same spot for months, move it from fully cold to fully hot several times and wait for the air temperature to change each time. This will help reset the blend door and vacuum control system.
Also, if the blend door actuator is in fact bad, that IS something you can easily replace. Just have a look at the new one so you'll know what you are looking for. By the way, they ain't cheap!
Hope this helps... Let me know what you find.