if you go to the home depot they sell some for insulation. it comes in various thicknesses, but i would guess that 1-1/2" would be good. it would give you enough to press the fabric in enough without coming through to the
back side. that stuff is like what packing peanuts are made of. you have to be careful not to split it when you press.
the real foamboard comes from an arts and crafts store and is thin. i don't think that's what was used.
the other usable material is foam that is sometimes used for mattresses. it's that grey stuff, very flexible. if you were going to frame it, that would work, too. it needs the frame to help keep it in shape, but i thank the stuff they used was closest to this and that's why a frame was needed.
if you do it without the kit, you could wrap the foam with the fabric and mount the completed project on a slightly smaller piece of masonite with strong glue.