If you paint a lot of diluted Elmer's glue on craft foam, the glue impregnates the foam, the water evaporates, and the remaining dry glue stiffens the craft foam. With enough coats of the glue it gets fairly rigid. That takes a few days of doing it though. The end result is it's a sturdy, hard-wearing material that can be damaged (visible creases) if it's bent over, and the flexibility is similar to Jango's plates in AotC (Look close when he's rolling down the platform on Kamino, it flexes)
My advice though is to check whether craft foam reacts to fiberglass resin on a small test piece, and if it doesn't, fiberglass the back. It's a lot less work than glue impregnating.
A footnote: You guys at the top should look at reinforced* craft foam for an allowable/recommended material for early era armor, since if you only give it one coat of paint**, it has a raw bone/leathery texture and can be "tooled" for engraving patterns, like leather, only a lot less expensive.
* Reinforced meaning backed with fiberglass (if it doesn't react badly) or glued onto a thinner than "armor on its own" piece of sintra or a sheet of aluminum duct plate (thin, light, and rigid)
** My wife's back armor took about 5 soaking dips in a bowl of diluted Elmer's before it was rigid enough, and 10 coats of paint before it was smooth enough. It was chosen as a workaround material because we were broke at the time and didn't have any plastic available, but we did have a LOT of craft foam, glue, and paint that was lying around for another project that I never got around to.
Final footnote: Even with all that said in favor of craft foam, I wouldn't recommend it for a full suit of Modern era armor (See early era above) because the many many man-hours of work it takes to get it anywhere near standards quality outweigh the few bucks it saves you over trash can armor. I'll call it an accent piece material or emergency stand-in at best. This final footnote is clearly the reason it's not usually allowable as a material.