stiff textile material for simple underbed "drawer"
PVC sheets may be a great option. You can use a thicker one that will hold it's shape better or a thin one that can be rolled up. Here is one of many suppliers just to give an example.

Personally, I'm a frugal and more about problem solving than how it looks and would probably use cardboard. I'd either use a bigger box from home delivery items or hunt down a large one, a household appliance store would have plenty of large and thick cardboard from such things as refrigerators. It's probably ok to just dumpster dive in the their recycle bin in the back but the best thing to do is ask up front.

Another option is plywood on furniture sliders and they come in square shape which would work very well too


pic credit: wayfair.com