You can buy this already done, if you look at SIPs, for glueing it up yourself, expanding foam does a great job as an adhesive, but don't apply the sheet immediately after it's squirted out of the can - let it sit a bit first to go gooey/tacky and acquire some structure otherwise it'll just collapse. It's also possible to get two part foams that form PIR when mixed, if you want to mix it up yourself, pour it into a mould that is holding both sheet sides and let it expand and set
Ply or MDF depends on the finish you want; if you're painting it use MDF and dont use a water based paint if you're not priming/sealing it. If you find a nice hardwood ply sheet with an appealing grain and want that as your finish.. then use ply!
I've got some insulated hatch doors made out of kitchen worktop for the decorative element