SMC,
Not special paint but good paint and primer made to cover vinyl. I started with the water closet to "experiment" in the least obvious area. I painted the walls a pale yellow and the ceiling a pale blue.
You have to take your time and really, I mean really get the surface ready to paint. The pattern on the vinyl will show through with just one coat of primer. You need to prime and let dry for 24 hours or more and then look at it. I only put one coat of primer on the water closet ceiling, because it's a little less textured or multi colored than the ceiling in the rest of the unit. I put two coats of primer on the vinyl on the walls, and I probably could have used three.
I am happy with the result but I did learn that painting over the vinyl will show ever nick or flaw in the wall or corners that are not mitered well. You can paint just about anything if you prepare the surface to be painted correctly.
It's not a one day project or even a weekend if you are willing to let the primer set up as it should.
Please let us know if you paint the ceiling. I keep looking up and thinking about it!