In this case, yes, it appears as though Zazzle has an official store for Alice Cooper products.
Some of the print on demand places do have "official" clients. For example, CafePress has Star Trek, Tupac, Dilbert, etc as "official" stores.
Here's a list of officially licensed images that Zazzle works with:
Create and buy personalized products using images from Disney, Star Wars, Smiley Central, Family Guy, Neopets, and more!
That being said...
You will likely find trademark/copyright infringement examples all over the web.
Especially at print on demand places that allow instant uploading of graphics. Since the upload is "instant", it's almost impossible to check whether the user followed their terms of service by uploading graphics that they own the rights to.
In those cases, the graphics may show up on the site until someone lets them know that they are there. Sometimes that "someone" is the original copyright/trademark holder in the form of a cease and desist (or lawsuit).
However, like the above examples, there are "official" stores on many of the Print on Demand places with officially licensed images.
Zazzle users somehow get to license logos? 
