Help with Receptacle Inside Cabinet
I just did this on the medicine cabinet I installed in my bathroom. The issue I think you are going to face is the depth of the wall - at least looking at the picture. I mounted a box on the side of the cabinet in mine but I had a full-depth wall. In your case I would tap into one of your 20 amp GFI protected outlets next to the cabinet using BX cable. Put a knockout in the side of the cabinet. Connect into the cabinet with a BX connector. For the receptacle I would use a narrow, grounded power strip mounting it on the back of the cabinet hard-wired with the BX cable.
I checked in my area and this met code when I used an outlet box. I'm not sure about hard-wiring a power strip but I'm sure one of the electrical experts here on Stack Exchange will educate us.
Edit

Here is one sample of a hard-wired power strip from Amazon. There are others available.