Good question. nantena points us to this thread over on Upper Deck's site that has the odds set from the Heroes of Azeroth loot cards at 1:96 packs for the tabard, 1:288 packs for the Hippogryph pet, and 1:566 packs for the turtle. That means you have to buy 566 packs, which is about 24 boxes (there were 24 packs per box in Heroes of Azeroth) at about $60 each, just to mathematically come close the odds you need. That's already $1,440, and even then, considering all the cards are randomly placed, you're not guaranteed a card.
Now, the latest set is a little easier-- March of the Legion has more cards per pack, and the odds on the loot cards are better. We don't know what they are, but you'd probably get at least one loot card (and probably more) if you shelled out $1,000 right away. However, if all you want is the loot card itself (you could get a nice tournament set out of $1,000, no doubt), you're better off checking prices on eBay. $500 for a Spectral Tiger seems like a lot (and trust me, it is-- if you're spending $500 on a virtual item, you have too much money), but it's nothing compared to what you'd have to spend trying to grab the card randomly.