Sharpie oil based markers work on plastic and can be used on plastic dolls, HOWEVER the ink will bleed deeply into the vinyl and over time spread out much father then you ever intended. Never ever use these markers for a doll's make-up. The only option I can see is using these is for complete recolors as the paint drys flexible, resistant to scratching, and does not get sticky or chip. The coverage is not very even though which makes a vinyl spray paint the best solution for complete doll repaints. I still have yet to find a supplier who carries more then the regular colors of vinyl paint, like white, black, red and tan. I contacted a custom auto painter and they can only get the paint in large bucket quantities, not in spray can form. :(
I have experimented using Testors model paint (intended for plastic models) with does work on the hard plastic bodies, but has a chemical reaction with the doll hair. (It transfers onto the hair and stays sticky.)
Most doll artists use acrylic paints or chalk pastels as makeup then top with clear sealers. This method is the safest solution for modification as it does not have long term plastic breakdown.
Here is a photo of a vinyl Blythe scalp that has been colored with a sharpie paint pen. You can see the flexibility and color cover of the ink. After removing the paint with acetone you can see the deep ink bleed. When the paint was removed the color bleed eventually faded away after 11 months. But results will vary based on the type of plastics and rubber.