We've used Polycarbonate as a substantial part of our robot construction for several years now, without any major issues.
Initially we used it only for the belly-pan of the robot to mount electronics on, but the past few years we've made drive bases, gearboxes, shooter frames, and more (We've also used it as a frame for a grabber arm, but I don't recommend this because it's not the most stable).
These systems work great for us and are quite a bit lighter than the aluminum-frame robots we used to build.
As far as thickness, we've used a variety ranging from 1/16" to 3/8" depending on the application. Keep in mind that thinner material can usually be made fairly ridged if supported in the right places (like the belly-pan in the first image above).