I have made that bag. Several of them, actually. For a single or double thickness of the webbing, my regular old sewing machine was fine. When I got to the top piece and the handles, I needed to switch to my industrial machine. It had no trouble with the webbing. I also make dog collars so I figured it would.
As for what type of industrial machine, I personally would recommend seeing if you can find an older model via craigslist or the like. I purchased a Singer industrial machine from the 1960s and I love it (mine's a model 188K). I call her Belinda.
I had looked at "heavy duty" singers in the local sewing shop, but the salesman even told me that Singers weren't made as well as they used to. Plus at $150 old Belinda was a steal. (Although she does sound like a truck and shakes the floor when I turn her on).