I got my webbing locally from a friend at a upholstery shop, but ssnake-oyl.com carries automotive seatbelt grade webbing as well, a bit expensive, however.
For stitching, I used nylon upholstery grade thread that I got from a local fabric store.
You will need a heavy duty, or a up to snuff older sewing machine, with a reverse stitch feature to sew the webbing together at the latch(s). I used my vintage Bernina 830 Record and it worked beautifully. I bouht my machine at a local thrift store for $94, which was a steal for this swiss made embroidering machine which has a more than powerful enough motor assembly to handle heavy duty and thick fabrics.