when i'm having a party and need to use less than wonderful looking chairs, i just throw a nice blanket or afghan over them. or you could by a chair slipcover.

You could try an adhesive like "Shoe Goo". CDO's suggestions would also work.

Trying to glue or otherwise mend this vinyl is useless. Quickest is the long way around. Go to a good fabric store, buy a matching piece. Unscrew the seat from the frame. Pull the staples. Throw away the old vinyl but keep the foam pad from underneath. Staple on the new vinyl and make sure there are no folds on the top side. Screw the seat back in place. Sit yourself down. Trip and all should take about 1/2 day if you have a decent stapler. It's a nice Saturday project because no real special skill is required. The vinyl goes on easily and a pair of sharp scissors, screwdriver, and a good staple gun will be all you need.

I would use liquid nails to glue it back under or Welbond.