You can find the in-game store right beside the dungeon journal and game menu buttons. I tested out the purchasing process myself and it was extremely easy (almost too easy -- my poor wallet). When I clicked to buy something a confirmation prompt appeared and let me know that it would be charged to the credit card I have on file with Blizzard. After confirming I wanted to buy, I had to enter my password to confirm, and not 5 seconds later the Cinder Kitten was sitting in my bags.
The only pets not available in the in-game store are the Guardian Cub and Moonkin Hatchling. This may have something to do with the unique nature of these pets (the Guardian Cub being BoE, and the Moonkin having different Horde and Alliance versions). This 50% off sale is also available on the web store and will end on January 1, 2014.
Patch 5.4.2 in-game store