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Answers
Turn the AC/heat knob back and forth a bunch of times and retest to see if maybe you got the flap to move.
Some of these systems work on vacuum so there may be a vacuum hose loose or leaking somewhere, affecting only the passenger side hvac flaps. Others work with mechanical cables. Others work digitally with solenoids & stuff like that.
An inexpensive hack that can partially mitigate the issue is to install a shutoff valve to the heater core so that you can turn the valve such that coolant does not circulate through the heater core during summer. This hack should eliminate the heated-air coming from that passenger vent - it won't be cooled either - it will be ambient temp.