Good quantities of World of Warcraft Companions don’t have a requirement for siding with a particular faction. However, some World of Warcraft pets are strictly attained through the Gods' favor, like the Green Proto Drake that you get from the Mysterious Egg. The companions which are solely attained by chance are called rare drops, and no matter how strategic your attempts are, sometimes you just cannot get your hands on the one you want without purchasing it from someone else.
There exist many WoW companions who are solely acquired simply through battling enemies. However, most are pretty rare, and it can seem heavily based randomness. If you try and try but nothing comes of it, there is nothing you can really do about changing their method for obtaining these pets other than to purchase one from another character. A large quantity of WoW players who can be considered well-known companion collectors have to repeat the same quests over and over again in order so they get the rare ones they need.
A rare pet known as the Azure Whelping can be attained by battling the Draconic Mageweavers, Blue Dragonspawn, Blue Scalebanes, and others. A player needs to travel over to Azshara so they can locate the mobs and possibly this uncommon companion. It is necessary to be skilled and well equipped in order to battle those mobs especially because they are all about level 50 to 54 elites. They have odds of roughly 1 in 1000. This randomness of being able to obtain it is just like that of Crimson Whelpings. Unlike Azure Whelpings in Azshara, Crimson Whelpings are discovered in the area of the Red Whelps.
In order to obtain the Ammen Vale Lashling, characters can join in the Argent Tournament. This even occurs in northeastern Icecrown and can be done at any time. Another perk of doing this is you can earn the Champion's Seals. Horde members may buy this companion through a neutral WoW Auction House. However, at times it is possible for them to buy them through their own auction house. Alliance members may buy race-specified companions and other corresponding Argent Tournament companions but only when they’ve successfully completed various quests.
In regards to Horde members, the Sprite Darter Hatchling is another drop determined by your faction grind and more easily obtained by them, as it solely can come from the Sprite Darters from Feralas which is friendly to the Alliance. A Sprite Darter Hatchling drops roughly 1 in every 1000 times. It is not impossible for Alliance characters to acquire it; Alliance members just need to do more level 48 quests.
The Green Proto Drake can be obtained in the Sholazar Basin. You must side with the Oracles and fight mobs from the Frenzyheart Tribe to receive the Mysterious Egg reward. It is not always guaranteed that you will get the Green Proto Drake from the Mysterious Egg, but it is certainly worth it doing the runs. You can farm this pet, but be prepared to put in some extra time and work.
One other example of a WoW companion that can be collected more simply, but is seemingly laborious to obtain is the WoW companion often referred to as a Phoenix Hatchling. The companion is known to have a 6.25% chance to drop from Kael'thas Sunstrider in the instance of the Magister's Terrace. This pet could be achieved in the regular and heroic versions of the instance.