How can I cut a vinyl chair mat to shorten it?
I've cut chair mats by laying them on a flat work surface. Then secure a metal straight edge along the cut line. (In my case I clamped a long piece of aluminum bar stock in place to the work surface with the mat in between).

A utility knife with a sharp blade was then used to score a cut line along the straight edge. Chair mats are a relatively soft plastic and it was not possible to use a score and break off type process. Instead I cut in the same place along the straight edge 3 or 4 times. I then removed the straight edge and clamps and positioned the cut line along the edge of the work surface. This allowed cut area to be opened slightly as the part to be removed was bent down over the edge of the work surface. Once the cut line was opened like this continued passes of the utility knife blade in the cut groove allowed the cut to be made deeper and deeper and eventually finish the cut.