Rick, I could've sworn you've made the conservative teenagers line before
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I don't recall teams making a base out of all custom molded fiberglass/carbon fiber; but I have seen some good nonmetallic basese. The Use of polycarbonate by 126/190 has already been mentioned; but some teams like 703 and 1126 use it alongside metal in some years. Rage 173 has also been known for the all Birch (pretty sure it's Birch) Plywood base for years.
A lot of teams last year attempted making ramps from carbon fiber; I don't recall how successful they were, but I don't know if that was the actuation of the ramp/lift or the material itself that didn't work.
Last year, one of my favorite teams, 1717 Made some very, very nice ramps from Foam covered in aluminum sheet.
I know a lot of teams have used composites for manipulators; Over the years several teams have used the stick of carbon fiber hockey sticks as a light COTS arm. Also, not sure if this meets your qualifications, but 330's arm last year was a nice example of mostly nonmetal construction; The arm itself was fiberglass pultrustion and the grabber was plywood on top with lexan on the bottom I believe.