The quartersawn white panel material is a unique sort of thing. It's about 3/16 of inch thick. I bought about 14 sheets (4ft x 5ft) of it from a guy who had is custom made for a wainscot he intended to install in his house. It's sort of like a hardboard core with a paper-ish veneer on one side (for equilibrium) and QSWO veneer on the other. When I bought it, I really didn't know what the heck I'd use it for, but I did know that I loved the look of QSWO. Since then, I've come up with some interesting applications for it. It works well for the panel portion of frame and panel style furniture. I'll be sorry when I run out of it.
Unfortunately, I can't recommend a place to purchase it, but I have come across sources for QSWO plywood in 1/4 size on internet, but the shipping kind of stunk. If you like the look of QWSO, but can't find it in your area or it's too costly, then consider using QWSO veneer over a red oak core. I've done this many times with great results. The edge of the red oak board is barely distinguishible from the veneer. I used this method on the Stickley Round Table project.
Veneering doesn't have to be complicated or require special equipment. Yellow glue spread in an even layer and equal and distributed clamping pressure - that's all there is to it.
I have bought great looking veneer from Certainly Hardwoods and Veneers (http://www.certainlywood.com/). Those guys are great, and have always delivered outstanding veneer.