I like my fabric purses and totes to be washable, so I use plastic canvas sheets found at thrift stores. They're flexible, but rigid enough, or it can be doubled. In the past, for similar projects: I've also used old worn out cutting mats, the 'travel' cutting boards (flexible plastic), inserts from supermarket bacon (well washed!), and $1 store plastic clip boards cut down with a router or dremal if I needed something perfectly rigid. The tops of broken plastic rubbermaid bins, or old food container lids. Another thought is to find the flexible plastic 3 ring notebooks at thrift stores...the ones where the covers are totally plastic and somewhat flexible. The regular notebooks are just plastic covered press board.. no good.
My husband has gone so far as to cut up plastic 2 litre bottles, flatten out the piece and tape to the stone floor, then turn the hair dryer on it until the heat get's it to lie flat on its own. He reshapes a lot of recycled things that way for his various boat and house projects.
Um...we like to repurpose and recycle where we can.