If you have a sign shop nearby go in and ask for their scraps. The corrugated plastic, like campaign signs are made from, is very lightweight but strong. I got an armload of various sized scraps for $10 and have been using it for almost a year (and I make lots of purses/bags). I've tried foam core, but it bends. Plastic canvas isn't strong enough to suit me.
I use an old rotary cutter to cut the plastic. Make fabric sleeves to put it in and leave one end open so it can be taken out when the bag is washed.