Huh. Well, lemme list off what I did, let me know where it stops responding.
Clicked 'Shop' at the top of the Launcher, then 'World of Warcraft' on the side, then 'Pets'. For me there's a row with three pets in it: the Moonkin Hatchling, Guardian Cub, and Alterac Brew Pub. The GC just says 'Learn More', the other two have prices.
Click on 'Learn More', it opens a page that says "Guardian Cub" in big headline letters with a button labelled "Continue to Battle.net Account" which opens a webpage in my regular browser. (I use Firefox almost exclusively, by the way. If you're using Internet Explorer, Chrome, Opera, etc, try a different browser?)
Once there, it may or may not ask you to log into your b.net account. It should tell you you can only redeem for one character on one WoW account . Pick your account (if you have more than one), then server and character. Click the blue Continue button at the bottom of the page. From there, the only thing left should be entering a payment type and clicking 'Confirm Purchase'.
Good luck--third time's the charm?