Sand the vinyl's surface with 220-grit sandpaper until the shine is gone and the surface looks dull. Work the sandpaper into the textured areas as much as possible, but if the grooves are deep, you may be limited in what you can do.
Wash the vinyl with trisodium phosphate, a sponge and scrub brush. Use the scrub brush to work the cleaner into the textured areas so it comes completely clean. Rinse the vinyl with plain water and dry thoroughly.
Apply a liquid deglosser to the vinyl using a soft rag. Buff it into the surface and then wipe off any excess with a dry rag.
Prime the area with a spray-on primer designed for vinyl. Work in a well-ventilated area on a covered surface. Spray the chair with short, even bursts, holding the can about 12 inches from the chair. Allow the primer to cure according to package instructions.
Spray a coat of paint on the chair in the same manner as you applied the primer. Use paint that is designed for vinyl. Allow the paint to cure. Apply a second coat.