I’m not actually planning to laser-cut this; I’ll be routing it on my OX. But if I had a laser, I’d be using it, and I know that that’s the general range of thickness for cutting wood with a 40W laser, so asking here…
What kinds of stores local to you do you go to in order to purchase thin plywood? I can’t get thinner than 5mm at the big box home improvement stores. I have some fairly simple shapes I need to cut, but no stock. 
Hobby / craft stores carry sections of thin ply. Places like Hobby Lobby or AC Moore. Typically get it in upto 12 x 24" sheets.
Forgot about AC Moore, thanks. My local Michael’s is currently out. (Ironic, I know!)
My local Hobby Lobby is most conveniently located on my way to/from church, but of course is closed on Sundays, and I guess I shouldn’t complain.
I just remembered there’s a Klingspor’s about 15 minutes away; they might have something too.
AC Moore had plenty, so I got a couple 12"x24" sheets and some basswood 3"x24" sheets. 
I have a couple of local hardwoods lumber yards that sell veneer in multiple thicknesses, which can be just lovely to work with. (It can also be miserable because the really thin stuff is hard to keep flat enough to stay in the focal range of the laser.)