5. and 4. Worg Pup and Smolderweb Hatchling -- You'll get these from quests given by Kibler, a goblin by the Horde flight point in Burning Steppes (He's neutral though, so Alliance can get these quests too). Both quests will send you into Lower Blackrock Spire to catch the pets yourself. You can actually drop off a couple ledges down to the bottom of the instance, and climb back up through some spiders to Halycon's den, giving you a perfect opportunity to grab both a spiderling and worg pup for the quest.
3. Mechanical Chicken -- This one is missed by a lot of people simply because it requires a few drops and a few annoying escort quests: Namely, you must complete the quests from Distress Beacons in Hinterlands, Tanaris, and Feralas. Once you do that, Oglethorp Ognosticus will provide you with An OOX of your own. Sure, it's a smaller one with no combat capabilities, but hey, nothing's perfect. The pain with this quest will probably be remembering which distress beacons you already grabbed while you were leveling up. The other thing about this quest is that the pet is bind on equip, so if you're lucky, you may just be able to grab it off the auction house.
2. Disgusting Oozeling -- This little guy comes from an Oozing Bag, which is a very rare random drop from just about any level 50 or higher slime-type monster in the old world. You'll probably want to just head to Felwood, Ungoro or Eastern Plaguelands and start slaughtering slimes. This one will take a while, especially since the Oozing Bag doesn't even always contain the pet, but when you do get it, he's at least a pretty unique little guy. He actually gives you an aura while he's out and drinks Dark Iron Ale.
1. Chicken -- If you're the type of person who thinks that making weird faces at animals is the height of comedy, this is the quest for you. To get this one, you'll need to head to Saldean's farm in Westfall and spam the /chicken emote at a chicken. Whether you finally learn proper chicken-speak or the chicken just gets annoyed at you is unknown, but eventually, the chicken will offer you a quest. When you buy some Special Chicken Feed from Farmer Saldean and feed it to the chicken, it will lay an egg, which you can then loot for your pet.
Now, as you might imagine, this quest is difficult for the Horde to do since they can't buy the feed (Plus, it looks like the Horde can no longer get the quest at all, according to some), but there is a solution. If you're on a PvE server (or know someone who can make alliance characters on your PvP server), you can have a friend (or your second account) create an Alliance character (even a level 1 can get the quest) and run it out to Saldean's farm. They can do the quest, and when the egg appears on the ground, you can loot it to grab the egg for yourself. You might even risk making the character yourself. If you logged your horde character out at the farm and log back on quickly enough, the egg may still be there.