I used 12 gauge vinyl and an industrial needle on my sewing machine. Also, I opted for thread that would stand out from the vinyl, while you could certainly try to find something that would blend in and not become a feature of your bag. This is not a project I would recommend for beginning sewers, as the vinyl can be a bit tricky to maneuver.
I chose to use a nylon corded handle because I find them to be pretty comfortable to carry around, but you could opt for canvas, or even make a matching handle with the vinyl itself.
Here is my original concept sketch if you would like to try out your own bag. Measurements are not included, as you can make this bag any size you wish.