You can paint it - but it's just going to look like painted vinyl, with chips and cracks, etc.

If you don't want to reupholster or slipcover, just toss a sheepskin over it:
http://www.ikea.com/us/en/catalog/products/60114112

I attempted to paint some vinyl chairs last summer using Krylon spray paint designed for plastic. I don't recommend it. The paint never seemed to "cure" and remained slightly tacky, until it absorbed a layer of cat hair. I ended up reupholstering them a couple weeks ago.

you could paint a pattern onto it, but I wouldn't recommend painting the entire thing a new colour. If you do decide to paint a pattern, use acrylic paint with fabric medium added to it. It's a bonding agent. Then you need to coat the painted parts ONLY with a clear waterbased varnish. My boyfriend paints on leather/vinyl shoes, purses and bags with this method and it's extremely durable.

Try a shoe dye, such as Angelus or Tarrago.

A good quality stain might do the trick.

Yes. There is stuff specifically for this. It's made by Plasti-kote and it's called Vinyl Dye :-)
I picked mine up at a Checker Auto Parts (a.k.a Shucks/Kragen)

http://www.plastikote.com/plastikote/auto/template.jsp?searchcode=RES&product=UltraVinyl

Is there something about those particular hammerhead tacks that you want to retain? If not, then I would think you'd get better results by making (or having made) a new cover in the color you want and then hammering in new tacks. A good upholsterer could even save and reuse those tacks if that was what you really wanted.

Or am I totally missing the point and you aren't even referring to the hammered tacks?

It's super cute the way it is, I thinks...

Thanks for the suggestions all! The vinyl just looks really dingy and I thought I could do something without having to completely reupholster. I really do love the existing hammerhead detail and I can't imagine trying to recreate that on my own. Maybe if I went the "embrace the problem" route and painted the legs instead then it might change the dingy look somehow? Or perhaps I should just leave it as is? I'll try the vinyl dye.

If you're happy with the color, you might just try giving it a really good cleaning - it's amazing how much dirt hides in the texture.

Agree with Dear Abby, maybe attack it with a Mr. Clean magic eraser first.

Thanks again everyone. I think I'll go with the good cleaning. The vinyl dye is a spray application...I'm not sure how that would work without spraying over the tacks.

those tacks are cheap and easy to apply. just look on the web for simple upholstery techniques.

I too think the tacks could be easily redone.