We used 1/8" polycarbonate sheet last year and it worked great. More commonly available at Home Depot, Lowes, and Menards than birch plywood, which can be somewhat of a specialty item. Polycarbonate sourced there is cheaper than the
sheets that AndyMark stocks.
I would avoid using construction grade SPF plywood in a 3/8" thickness, as its quality varies widely and won't hold up as well as birch, which has more glue in it.
You'll find that acrylic sheet is also available and cheaper, but you shouldn't use it. It breaks too easily, and the broken pieces are quite sharp.