For the puzzle to work, we'll need to hide the micro NFC/RFID tags in the bases of the pieces you'd like to use in your puzzle. The most straightforward way is to use felt-bottomed pieces. You can carefully peel off the felt partway, insert the tag and close it back up.
You can use a little bit of super glue to seal the felt back onto the base.
Here are three finished pieces, as well as an NFC ring. The ring contains two tags in it, on in the "front" and one in the "back" (relative to a small jewel embedded in it). This can be worn by a magician or escape room operator and is used as a secret tag to discreetly reset the drawer when needed. You could also designate other pieces or artifacts for this purpose.
Mounting the reader and electronics under the chess board is pretty straightforward. You can use very small screws or even double-sided foam tape to mount the reader flush to the board.
Place the reader's antenna coil under the two board spaces you'll use in your puzzle, in this case I'm using g3 and h3.
Depending on how you'll be presenting your board for the puzzle, you may need to cover the bottom to avoid spoiling the gag. I used a piece of baltic birch plywood stained on one side. I glued in corner braces to fit and hold the bottom in place.
You can place felt protective feet on the corners for a nicer finish and to protect the table beneath. If you haven't got any on hand, you can improvise with some felt, a hole punch, and some glue!
Who would suspect anything is out of the ordinary?!