I've got a simple White portable sewing machine, and I've sewn up 4 different tents, and many canvas bags with it - including attaching lots of webbing. I use a Sears #18 needle - but most any heavy-duty needle will work. I also use cotton-wrapped polyester for thread. The polyester core gives it strength, and the cotton wrap matches in with the cotton canvas.
With webbing, you do have to do something to stop the ends from unravelling. Like was mentioned before, I als will use my machine and sew a number of zigzag stitches along the ends, or severa stitch several lines across or parallel on the webbing - anything to keep the cords in the webbing from pulling out. I have also lapped cloth or Leather around the end of webbing and stitched it in place. That leather also adds its own extra strenght to the end of a strap.
When sewing through multiple layers of canvas and/or webbing with my portable sewing machine, I often turn it slowly by hand instead of using the electric motor. It's much easier to control it that way, and adjust to how well it is stitching through the material.
I've even sewn some light leather with my machine, but only occasionally. But so far I haven't tried sewing carpet - for when I am sewing up traditional carpetbags. It's just too thick to feed through. So those I just stitch by hand.
Here's a couple pics of the last carpetbag I made. New carpet, canvas lined, wood bottom inserted, metal folding rim, leather trimmed, and leather covered rope handles. It is made in the style of carpetbags from the mid to late 1800's. Yes, I'm a Yankee Carpetbagger!
Just my humble thoughts to share. Take them as such.
Mikey - yee ol' grumpy blacksmith out in the Hinterlands