You wouldn't need to laminate any materials together but you would need to cut the parts separately. The board with the shapes cut out could be done with something like 9 or 12 mm MDF or ply machined to a depth of say 6mm then the figures cut from a sheet of 6mm material so they were flush when put in place. Then or course you would need a printer to print the images for the board & animals.
Cad software is what I would call a design program so that plus cam software like cambam or easycam to produce g code & them mach3 or what ever to run the machine.