Thin, Flat, Rigid Sheets? (cheap)
How about a very thin acrylic sheet/plexiglass. It may not be as rigid as you need (being too thin), you have to check that for yourself. Also, plexiglass is not a very cheap material, generally speaking, but the price depends on the size and the thickness of the sheet, so what you are looking for may not end up being expensive. If that doesn't work for you, I'd suggest you go and look around a place where they sell construction materials and you will likely come up with a good solution.