A 100% acrylic paint will adhere directly to either fiberglass or vinyl. An acrylic based primer can also be used first. What does not do well on vinyl is an oil paint. Oils are very brittle and inelastic. Vinyl expands and contracts. Bad combination!

To get the edge which shows to the interior when the door is open, requires that the door be removed from its track.

I would not say that this is a great idea, but it is possible. The brand of the door might matter if you have vinyl rubbing on vinyl when you are opening and closing the door, paint will not stick there, or if vinyl touches vinyl when the door is closed, the paint may not stick there. Also, where the little lever is that you move up and down to lock the door sometimes that will rub on the vinyl, so the paint will rub off there. These doors are not always designed like wood doors that the manufacturer knows will be painted so this may create some maintenenance and touching up now and then.

Thanks for the responses. I'm torn. I really dislike the current color, but this could end up being an even uglier disaster (chipped paint). The home inspector said he thinks it's fiberglass. How would I go about finding what the actual material is?

I forgot to mention one VERY IMPORTANT caveat about painting vinyl: it is imperative that the new color not be substantially darker than the original. Vinyl expands when the heat of the sun shines on it. If painted dark, it can literally permanently warp and destroy the door! This is not the case with fiberglass, as the glass fibers controls expansion. Of course, this is especially important if on the south and west sides of the home which get the full brunt of the sun.

Painting a white door to a creamy white or light tan should not be a problem, but never anything darker! Avoid grays, as the black pigmentation attracts heat.