How can I mount an electrical box to exterior foam board?
That is called fan-fold insulation. Typical for under vinyl siding in older homes.

Very surprising that there is no sheathing, but not unheard of. What I would do is get a round cut-in style fixture box and mount it directly to the vinyl block. This seems like your only option, apart from removing some siding and replacing the vinyl block with a fixture box style piece from Arlington Industries.  



or



http://www.aifittings.com/catalog/siding-mounting-blocks/siding-mounting-block/