Square or round the tear with scissors to get the edges even. Push the cut scraps into the squared or rounded hole to fill it. Regardless of the damage, you'll need to even the tear or hole out to patch it properly.
Place the vinyl patch over the filled rounded or squared hole to measure the cut size and to compare color contrasts.
Cut the vinyl patch a larger circumference than the rounded or squared hole. Place over the tear or hole once again to check that the cut size will cover the tear or hole and surrounding edges.
Dye the patch if necessary. Apply coloring dye to the patch until it matches the damaged chair. This might require a bit of trial-and-error, following the dye manufacturer's instructions and testing several patches until the right balance is found.
Apply a thin layer of adhesive to the edges on the back of the vinyl patch with an applicator. Do not apply adhesive to the middle of the vinyl patch or it will stick to the cuts stuffed into the tear. Press the vinyl patch into place and let dry for at least 24 hours before sitting in the vinyl chair.