While the characters do seem to have a fan-designed set of personality traits, the songs and music videos made with or based on the programs usually have unrelated story lines. Personally, I kind of like it that way. You never really know what you'll find when you go looking for new Vocaloid songs.
The closet thing to an "anime" I've seen is Black Rock Shooter, but that's not even Vocaloid. There was just a Miku song dedicated to it, and the main character vaguely looks like Miku. The similarities end there.
There is, however, already a short Vocaloid manga. ("Hatsune Mix") Its chapters are are series of disjointed oneshots, too. It's kind of cute. I liked it.