First off - this is located in Seattle and isn't conducive to shipping.
The shop I work at is preparing to move facilities and we have a lot of scraps that are likely going to be trashed. We have a decent bit of finished plywood (lot's of 3/4", some 1/2" and 1/4"). Some larger 1/4" raw poplar plywood pieces. Apple ply (also known as eurocore - high-end, super sturdy plywood) in 1/2" and 3/4" sizes.
We also have a large amount of scrap Richlite
- this stuff is awesome (and expensive) for countertops, cutting boards, camp tables, whatever surface you can think of. It's made from recycled paper and is waterproof. Only downside is that it's rock hard, so it will dull knives (I still use it as a cutting board and just sharpen the knives more often). Here are some photos (photos flipped 90 degrees, sorry). We have it in black and brown (a few pieces in "blonde" brown).
If interested let me know what you are looking for, rough dimensions, etc. I can't guarantee anything, but I'll do my best to check and snag what you're looking for... just hoping to pass along some of this great material to others before it gets trashed. If looking for any other woodworking material I may be able to scrounge additional items, just let me know what you need.