I used a regular sewing machine. It worked very well.
I cut the vinyl the thickness of the side panel oversized for the inside piece (the panel with foam on both sides runs around 1.5" thick before the foam is compressed). It doesn't need to be stretched, just snug. The outer piece I made double that but didn't really need it. Since you can trim as you go it doesn't matter if it's too big. The ends with the tack strips will stretch an extra 3/4" on each end.
The seam allowance at the top was just half the width of the sewing machines foot. If you cut 1.5" around the side panel, then at the top, under the arm rest, you will lose perhaps 3/8" of that to the seam allowance. That leaves 1.125 on each piece of fabric (totalling 2.25") to go over the top of the panel. Since the panel is 1.5" then you will gain some length at the bottom.
One hint I can offer is that you should check which way your vinyl stretches most readily when it's normal temperature. Mine only had good stretch in one direction and you want to make sure it stretches from the front of the chair to the back.