I bought some 1/16" 3 ply sourced from the jungle site for the laser. Typical cheap stuff, country of origin unknown. It looked OK initially, then warped sitting on the table a few days after unwrapping. The plies separated fairly easily in areas, but I've only used one sheet so far. Smaller parts would probably be fine after gluing, but I'm not sure I trust it for larger structural parts. Of course, your results may vary as one order or vendor may have substantially varied quality.
Searching google for 1/32 seemed to return a fair price for Midwest ply at Menards. Unfortunately, it looks like they're discontinuing it. It didn't come up in my searches when I ended up buying from scamazon. I'll have to stop there today to see what they have.
I didn't realize there was an Aircraft Spruce warehouse so close to here. Will try them next time.