Directly asking readers for fanart through a blog or social media is a great way to get fan art. I've run several fairly successful campaigns over the years, and I've gotten tons of great stuff from awesome contributors.
Here's some fanart tips:
1) Wherever possible, give lots of notice. People lead busy lives and need time to put something together.
2) Tell them what their art will be used for, ie. filler, vacation, holidays, exams, milestone, you just had kids, etc. For most people, knowing helps.
3) If possible, promote the fan art when it's convenient to the maker. Sometimes they could use an extra boost in traffic. A strategically-timed pimp can help them out when they need it most.
4) Be sure to give proper credit. Fan art makers seldom think to include a name/URL/date on their image. I add a tag to the bottom of each of my fanarts, and I typically ask what handle the artist wishes to go by.
5) Have a sincere appreciation for every single fan art you get. People have gone out of their way to make something koo for you. So thank them, pimp that shit hardcore, send them some traffic, and if you really like the person, consider doing a reciprocal fanart or send them a couple prints or something.