You should be able to get a vinyl repair kit at your auto supply store. It is used to repair vinyl seats and roofs (like when the garage door opener falls off the ceiling and scratches the vinyl roof of the car - could only happen to me :o)
I recovered my bean bags by cutting pieces a little larger than those on the chair and sewing them together. I made a hem at the botton and threaded a piece of nylon rope through the hem to use as a draw-string. I used a fur material; anything with some strength to it would work. That way I didn't have patched chairs but "brand new"!
You can purchase different colors of tape or use clear. Your problem with the corners coming up is that it should be cut with rounded corners.
What about an iron patch, the kind used to patch clothes. I would apply it to the inside of the vinyl. Turn the bag inside out and cut the patch @ 2" bigger than the tear, make sure the spot to be patched is nice and flat. Put a dish cloth over the patch and press with a medium hot iron. You could keep the duct tape on the outside it will help to keep the tear flat and remove after the patch has cooled.
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