What are LEGO baseplates made from?
LEGO baseplates are made from ABS plastic, like most of the opaque rigid plastic LEGO elements since 1970. ABS is a thermoplastic consisting of a styrene monomer with the addition of an acrylonitrile momomer and a butadiene monomer. There is no PVC or vinyl present.

I suggest reviewing this thread for more on laser-cutting ABS. It is not a trivial matter to get a good result.

Edited to add:
In response to a query from a "safe toy" website, LEGO categorically stated that they do not use PVC in any of their products except in rare instances like electrical wiring insulation.