Recovering Chairs With Vinyl

Try cutting a triangle shaped wedge out of the corners of the plastic. Cut it just deep enough so that it covers the seat, and the overlap the corners and staple. I suggest making a newspaper pattern first and seeing how it works before cutting the vinyl.
Also, are you heating the vinyl slightly to make it more pliable? This could be done by using a hair dryer to heat the part that you are working with. Practice with some scraps before actually upholstering.