Thanks for your question and welcome to the community.
This is one of those situations where you actually have your answers for you on the links you've given.
Especially on the Instructables website; everything you need for this build is there.
My main advice would be to go over the DIY Lightboard Materials and Tools List on the link, everything we sell save for maybe the markers we sell at your local Home Depot store.
Be aware that if you use glass, we no longer cut it as a service. Same goes for plexiglass as well. We do provide basic cuts for wood, but not for cutting the channel to allow the LED lighting to rest into the frames.
And since you didn't give us direction with what materials for the frame (wood or metal) or specific sizes, you'll need to do this before you come into the store.
These materials are found in various departments throughout the store, so you'll need to speak to a lumber/building materials, lighting, and Millworks associate who can get the requested parts for you.
As for the LED light itself, we carry a brand by Armacost that requires a separate power supply transformer, but it works well due its extremely thin design.
Again, and I really cannot stress this enough......your materials and tools list is already laid out for you on the Instructables link. Use that to come into your local Home Depot store.
Know the size and makeup of your build, and then come into the store, and we will be more than happy to assist you grabbing these supplies.
To mimic the technical end on the YouTube link you gave us, use a quality camera/tripod as well as a mirror to convey the board in reverse.
Let us know if you have any additional questions,