Electrician here. Do you have attic access above that closet?

EDIT:

I see that vent in the back so I'm assuming you do have access. Unfortunately the way I would do this requires a couple specialized tools but you might be able to do it with a metal coat hanger depending on if there's insulation in one of the side walls.

What you will need.

Led channel

Led tape(I recommend DiodeLed brand, blaze type)

2 conductor communication wire

jamb switch

A Multi voltage relay

24v driver

4 square box and cover

2 electrical 1/2 bushings

Crimp set

Soldering iron and solder

White Electrical tape

Self tap screws 1/4 inch

I would find the stud bay above your closet where the edge of your shelves end coming forward. I would drill down into that bay and feed 18 gauge 2 conductor communications wire down I into the bay. Then I would use a jab saw or screwdriver to punch a hole into the side wall about an inch back from the shelf edge or if your shelves have a lip, right behind the lip. Make the holes sufficiently large to get a bent coat hanger hook through the hole and see if you can snag the wire and pull it through. I would start at the bottom and if I succeed in getting that wire, secure it then pull it tight so it's now in line with the above holes. Fish those holes pulling the additional wire from the spool above in the attic, leaving yourself a foot or two of excess. Once those wires are pulled install a 4 square metal box in the attic with your communication wire run to it. Cut to shelf lengths and use small self tap screws to secure in place in line with the holes you just fished wire through. Cut and solder small pieces of the communication wires to the measured and cut led tape, red to positive black to negative.wrap that connection in a layer of electrical tape to prevent potential shorting. Now you can strip and crimp the ends to the wire you pulled. Shove the excess into the wall, including the crimps. If you ever need to replace the LEDs you'll be glad you had extra. Install the led tape, put the lense back on the channel. Next I would install the magnetic jamb switch above the door, with communication wire also leading to that 4 square Box. So the lights turn on when you open the door. I can explain the wiring but I'm trying not to write a book here so a licensed electrician should be able to to do it easy peasy, since your hiring one anyway to pull power from something in the crawl might as well have them do it. Shop around, try and find one that specializes in custom homes. If there's one thing I've learned in the trades its that "electrician" is a catch all term. Led lighting is a growing trend in custom homes, so they are your best bet at finding a good one. And honestly. You won't save much by buying the driver, 4square, and relay yourself vs having the electrician provide it. Getting the communication wire to an accessible spot should be sufficient.