Do note that the credits mostly go to other people, all i did was post here what i found.

Cait Sith Doll
Pre-Order Bonus.

Baby Behemoth
2.0 CE Bonus

Mammet #003L Mammet #003G Mammet #003U
Reward for completing their respectives companies Hunting Log

Storm Hatchling Serpent Hatchling Flame Hatchling
bought for 20,000 seals from your company shop.

Morbol Seedling
Second Chest in aurum vale

Tiny Rat
Purchased from a vendor in highbridge for 2400 gil after completing the FATE "Attack on Highbridge"

Baby Bun
Get maximum reward at the FATE "Lazy for you"

Chigoe Larva
Reward for completing the sidequest "Occupational hazards"(from an npc west of quarrymill)

Bluebird
Unknown

Wide-eyed Fawn
Purchase from Auriana for 125 allagan tomestones of philosophy in Mor Dhona

Infant Imp
Get maximum reward at the FATE "Go, Go, Gorgimera!" in Northern Thanalan

Gravel Golem
Craft by goldsmith, level 46-50 recipe.

Tiny Tortoise
Gathered by fisher, VERY rare catch in salt strand in lower la noscea(use Northern Krill, Floating Minnow or Yumizuno)

Baby Raptor
Purchase from boughberry trader in the Redhive for 2400 gil after completing of FATE "Clearing the Hive"

Baby Bat
Purchased from junkmonger Nonoroon for 2400 gil in Poor maid's hill, the NPC "nonoroon" appears after completing of the "Poor Maid" FATE in the same place, slowly walking and setting shop, just follow until its settled.

Dust Bunny
Purchase from Auriana for 125 allagan tomestones of philosophy in Mor Dhona

Wind-up Dullahan
Craft by armorer, recipe of level 46-50

Gigantpole
Rare fishing reward at level 50 from The Burning wall using Tapwater frog bait.

Pudgy Puk
Get max reward in the FATE "The eyes have it", west of camp dragonhead.

Buffalo Calf
Unknown

Cactuar Cutting
Reward for 'Zombies Are People Too' - Level 47 quest starting in Southern Thanalan(24,40). NPC's name is Hab.

Smallshell
Get max reward in the FATE "It's not lupus", south of costa del sol

Beady Eye
Reward for 90 ininterrupt days of sub to the game.

Fledgling Dodo
Purchase from Auriana for 125 allagan tomestones of philosophy in Mor Dhona

Coblyn Larva
Gathered by miner, level 46-50, Northern Thanalan, Blue Fog

Wind-up Aldgoat
Crafted by weaver, level 46-50 recipe.

Fledgling Apkallu
Unknown

Goobbue Sproutling
Reward for main storyline quest

Bite-sized Pudding
Obtained from last chest in wanderer's palace(level 50 dungeon with the tonberries iirc)

Model Vanguard
Crafted by goldsmith, level 46-50 recipe.

Demon Brick
Inside a chest in Amdapor Keep

Tender Lamb
Unknown

Kidragora
Gathered by botanist, level 46-50 spot.

Wind-up Goblin
Unknown

Wind-up Cursor
Reward for 30 days of sub to the game

Wind-up Airship
Main storyline quest reward

Wind-up Qiqirn
Crafted by weaver, level 46-50 recipe.

Black Chocobo Chick
Reward for 60 days of ininterrupt sub to the game

Wind-up Dalamud
Reward for pre-order of the "Before the meteor OST"

Wayward Hatchling Mammet #001 Cherry Bomb
Bought from black rabbit trader for 2400 gild