3/16" is most likely rigid enough, so you might give it a try. The reason why most manufacturers use 1/4" is because they have it in stock, and use it for smaller base plates, (where they figure 1/4" is the minimum to not sag), and that it is thick enough for the mounting screw heads to be sunk slightly below the surface, so they don't catch on the work. So, if the screw heads that mount the plate to the router will fit, or countersink, into 3/16", then I'd try it out.
There's a sign maker on eBay where I've bought a lot of mine, and he will custom cut it for you. I can't remember the guys name right off, but he handles a lot of it. It would be listed under a search for Plexiglas, Plexiglass, or acrylic sheet. If you don't want it clear, and can get by with black, check out the plastic Seaboard material, as you can pick it up rather cheap. Its tough stuff, and they use it on the exterior portions of large boats, etc. As a matter of fact, I'm going to use it for the top on the next router table I build.