For some reason, I recently stumbled across a three-year-old hackaday article linking to a n-e-r-v-o-u-s blog entry which @HalfNormal previously linked back in the G+ days:
Following that path ultimately brought me to
Thanks, I think, to n-e-r-v-o-u-s, they now advertise selling plywood specifically for laser cutting:
We also offer a 3 ply plywood using melamine glue for laser cutting. A one sheet minimum purchase is required and can be cut to 24" x 48" for UPS shipping. Most common thicknesses offered are 3/32", 1/8" and 3/16" with birch faces and basswood cross band. Other species are available.