I'm not sure specifically about shadowboxes, but there have been threads about back lighting an object with LED. I know a few folks on here that have done it, including one of our fellow New England PPFA members that used it in a competition a few years back. In their case, they were using sheets that contained LED's (similar to a laptop screen).
There was also a vendor a couple years ago, at the trade shows, selling such a product. They may have been there this year, too, but I didn't see them. (nor do I remember the name)
We had our kitchen re-done a few months ago, and opted for LED under the counter lights. The lights are super bright, and are self adhesive. (tape) They are on 16 foot rolls, and require a 12v transformer. The ones they used can be cut every inch or so. There are 36 LED's per foot, and they give off very little heat.
Something like this would probably be ideal for such an application, too. The rabbet would hide it. The only concern would be if it is TOO bright. Then again, they work fine with a dimmer.
We opted for LED's that are 2700 kelvins. (the same yellow light as an old screw in incandescent) Some others are very white or very blue.
We have only used them in our kitchen, so I can't be sure of use for this application. In any case, here are some sources/pics/videos: (and maybe other ideas on the same site, perhaps not as bright/costly)
http://www.illumenate.com/dlblaze.htm
http://www.amazon.com/Diode-LED-DI-0090-16-4ft-performance/dp/B009ERBCC0
http://www.bulbamerica.com/diode-le...-2700k-high-performance-luxury-led-strip.html
http://www.businesslights.com/diode...net-strip-light-2700k-12-volt-16-4-spool.html
Hopefully you find more results searching the other threads, and someone chimes in with additional suggestions.
Mike
PS: Here is a photo of a short area of them in our kitchen. Theyre smaller than the width of a pencil.
https://www.facebook.com/photo.php?...604.1073741826.100000871517210&type=3&theater