If the whites of your eyes...

I don't think so. The light goes through the hole in the front and hits a nerve at the back. Then your brain interprets the signal. So I don't think the colour of your eyeball makes a difference?

But I'm no expert on this.

I'm not an ophthalmologist, but I think vision is a function of light passing through the lens of the pupil to the optic nerve; the "whites" are just the vitreous humor. I'm not sure how that could turn black without it indicating something really bad (there's no pigmentation in there), but I don't think it would directly affect vision.

I think lots of other animals have a pigmented sclera, so it probably wouldn't have much of an effect. (Except on other people.)

I saw a show where two inmates had tattooed the whites of their eyes different color. I think it was a gang thing, but I don't remember. One guy tat'd black and one guy tat'd red. They looked really weird. The guy with the red eyes was out and about and was making money tattooing other people's whites various colors.

From what I recall of the show (you might find some part of it on Youtube), their vision was not effected.

Here is a link to the diary of someone who got the whites of their eyes tattooed black. Apparently the vision is fine. Tattooing of the sclera is becoming more of a thing in body modification circles and I don't think the coloring itself would affect vision (though in the case of tattooing there are all sorts of risks of inflammation and infection and such, but I assume that's not an issue in your story). You can probably find further information if you Google "eye tattoo," "eyeball tattoo" or "scleral tattoo."

To add to the above, you can buy coloured contact lenses which completely change how your eyes look, and just have a clear circle in the centre, so no, I don't think it would be a problem from a vision point of view. Certainly nothing that I can remember from my biology degree suggests that the whites of your eyes have anything to do with vision. Also, certain illnesses can change the colour of the whites of your eyes - jaundice can turn them yellow for example.

in medieval Russia it was fashionable for noble ladies to blacken their teeth and the whites of their eyes. Don't know what the process was but it must've looked horrifying.

No impact on vision. You see through the hole in the middle of the eye with the light-sensitive cells at the back of the eyeball - the color of the sclera doesn't matter at all.

(For reference, plenty of animals, inc. our close cousins the chimps, have dark-colored sclera. Our white is a cosmetic oddity.)

So long as the cornea, the clear anterior layer of the eye (and of course the pupil, lens, retina and all that inside) remains normal, then it shouldn't be an issue. The cornea not only admits light through the front of eye and into the pupil, but it refracts it. So the entire cornea would have to be unaffected for the vision to stay normal.

http://www.selectspecs.com/info/structure-of-the-eye/