Google Play Music has a built-in album art grabber, I think. I know on my computer, Play Music has not only cover art, but even photos of the bands and things like that. I don't usually use Play Music anyway (don't like the way they organize things), but it would surprise me if it's not downloading art on its own. Check your settings -- many apps will not download cover art without your approval because they don't want to waste your data connection.
If I may, I'll plug my favorite music player app, MixZing, which has a built-in cover art downloader. I prefer it over Google's app because it organizes multiple albums by album, then track number; as opposed to track number, then alphabetical.
(And no, I have no connection with the people who make MixZing. I just like their app.)
--I do a good impression of myself--