What kind of paint marker will work for flexible vinyl?
I would try with a sharpie or other alcohol based marker. Look for the industrial version. 

As I wasn’t aware you needed colors, I didn’t recommend posca paint markers, which are acrylic-based and permanent. 

What it sounds like you’re going to need to do is to clean the surface to be painted with rubbing alcohol. The apply a fine mist of plastic primer from a spray can in several light coats. You can usually find them in most automotive shops. Then apply your marker and let dry. Cover with a gloss coat of spray varnish.

One last alternative might be colored nail polish.