I'm not exactly sure what type of bag he has, but for the bag I made for my Hikaru cosplay, I used a combination of fabric, cardboard, and some thick vinyl we had laying around. The vinyl made it thick enough so it wasn't flimsy, but also kept it from looking too hard with just the cardboard. I still needed the cardboard, though, because it still wasn't quite stiff enough.
It was a lot of layers to sew through, but it turned out great and I'm still getting a lot of use out it. And it was really nice having it as a part of the costume; always had a place to hold stuff. Not sure if that helps or not, but good luck with that!
Costumes I'd like to make in the near future:
Chris (Obstinate Arrow uniform), Scrapped Princess - 75% done
Robin V/Damian Wayne, Batman comics
Prince Alexander, King's Quest VI - 75% done