You can drill/punch a hole in foam core for mounting. Use wide washers (fender washers) on either side of the foam core. The wide washers will compress against an larger area of the foam core than just a bolt head and nut.
It certainly depends a lot on the size and type of robot, but I have seen a flat robot base made of a double layer of foam core. Motors were mounted using wide washers. The robot was made with foam core, rather than plastic and metal and gained rigidity with foam core cross sections. It was a light robot. Assembly was done with hot glue and tape.