There are a number of companies that make black foamcore but, as Jeanette said, they only make 3/16" as far as I know.
If you really really have to have 1/8" black foamcore and nothing else will work, you can make do by placing full sheet of 3/16" black foamcore in a heat/vacuum press at about 160 degrees for about 2 minutes and you will get a close approximation to a sheet of 1/8" black foamcore. The combination of heat and vacuum will soften the foam center and constrict the board to approximately 1/8". You might want to put some 1" wide strips of 1/8" scrap foamcore around the edges of the board so the bladder of the press doesn't round off the edges of the board.
Really.
Try it.