I just received a shipment of wood that I bought on eBay. It was listed as baltic birch plywood, 1/8" thick (3-ply) by 12" by 22", 50 pieces, good both sides. I went halvsies with another member of the Aiken Woodcarvers club. The total cost with shipping came to $83.00. If I cut each piece into two pieces, that gives me 50 pieces of 11" by 12" (or smaller, like 8.5 x 11 or 8 x 10, which fits cheap frames from Hobby Lobby).
This comes to 83 cents per piece or 42 cents per side if I cut them all up, and the possibility of using some for larger works. If I ruin any, I can always use the other side.
The birch facing is much better quality than the typical wood of this type available in hobby stores. There are always a few blemishes on the surface of this wood, but "good both sides" means that there are no patches that mend knothole defects, and the baltic birch plywood has a solid core. In the 1/8" thickness there are only three plies, but that is no problem.
There seem to be several suppliers that are continuously present on eBay, and other sizes and thicknesses are available. If a bulk order of this size is too big for you, you might be able to split the cost with a friend. Don't forget scrollsawyers too. They may prefer 1/4" thickness; that would be OK for you, but cost a bit more. Also, don't forget to provide positive feedback on eBay to the supplier.