I would bounce it off the AHJ first. The problem I see is that cords getting shut in the door. If it is a subsitute for the required receptacle per NEC that I would discourage it.
The code required recepticle will be about 5 feet to the right and below the cabinet.
IMO it is far more likely for someone to plug an appliance into that recepticle and then put it into the cabinet still plugged in (thus closing the chord in the door) than vice-versa also I can't see a 2-oz plastic cabinet door with a week magnetic clasp damaging any chord
Also, I've seen the aftermath of dropping a straghtening iron onto a carpeted floor (fire). However this recept. will actually be less convenient to plug into than the one right next to the sink (the med cabinet is to the left of the vanity mirror, and the recepticle will be facing the wrong direction). plus fires CAN occur in a lot of ways, I doubt a poorly insulated cabinet will be able to build up enough heat to spontaniously combust, sure the fire will be worse if it DOES occur, but a lot of people have wooden shelves in their bathrooms where a fire would be even worse.
as for installation, this recpticle will be fed by a single piece of #12 NM, with slack added and supported by a stacker, to allow easy installation and removal of the box, and a single #12 pair straight to the back of a recepticle will need a ton of troobleshooting down the line.
Of course the AHJ will be the final word
-Will
