There's lots of materials available as flat thin pieces. The question is, are they suitable for pickguards by their properties. The other question is, do they look good. And the third question is, do they fall into the aforementioned category of "plastic". I've seen fabric, napkins and drawings molded into clear epoxy, but isn't that plastic as well? There's nothing wrong with plastic, after metals it's the most tolerant material against pick scratches.
You mentioned Masonite which is basically cardboard. Laminate used for tabletops and floors is printed paper and some sort of epoxy flattened under high pressure which makes it very wear resistant. Any wood based material like chipboard or plywood would sooner or later wear out without a tough finish (=plastic).
Stone based boards are heavy and very brittle especially in the required thickness so plasterboard or fibre cement are out of the question. I guess ceramic would fall into that category. Glass comes in sheets and can be shaped, doesn't scratch easily but is not shock resistant. Zirconia used for knives would be hard and thin but where could you find it and how would you shape it? Same goes for carbon ceramic used in brake disks. -For what I could quickly learn it seems that ceramic would make a very cool pickguard. The biggest issue with it (outside pottery) is that it can't be made at home.
Natural materials other than wood might be interesting. Leather was already mentioned. Turtle shell would look nice and even have some guitar related history as picks but it's not environmentally sound. But how about cow horn? It's hollow so there's lots of surface when rolled flat. And it's tough, buffs shiny etc. The only problem with that is to find instructions about preparing the horns so that they won't roll back or split. There used to be a comb maker in an outdoor museum when I was a kid but when I took my kids there they told that the skill died with the old master.