I would test of the backlight shines through two layers of the vinyl.
If it does not
- buy enough vinyl to do the letters twice
- fold the vinyl in the middle, double-tape the insides together
- cut the letters out
- now you can put one set of letters on top of the plexi and the other set on the back.
Or, if the vinyl is thin enough, just double-tape two layers of vinyl together, cut the letters out and put them on top of the plexi.
If the light shines through two layers of vinyl
- double-tape the vinyl to a piece of thin cardboard
- cut the letters out
- put the vinyl letters on top of the plexi
- double-tape or glue the cardboard letters on the back of the plexi.
Finally, like big10p said, put a second piece of plexi over the top of it.