Since I had to look it up recently, here's the list of quests required to unlock the Moogle beast tribe quests:
The final unlock quest is Tricks and Stones and is located just outside of Moghome (The Churning Mists - X: 27.2, Y: 34.5). The Main Story Quest has to be progressed through past He Who Would Not Be Denied - some of the quest chain quests are available before that but the quest chain itself requires it before it can be completed. Flight within the Churning Mist is also required during the unlock quest chain. (Weirdly Tricks and Stones indicates it unlocks after the Aery but the unlock sidequest requires the MSQ be progressed past that.)
The first quest chain is the Pomguard quest chain, and it jumps around a fair bit and frequently resumes nowhere near where it ended. (This is why I've included the locations.) Flying is conceptually not required for this but recommended for your sanity.
- A Pebble for Your Thoughts (The Churning Mists - X: 29.7, Y: 35.3)
- Spineless Wadjets (The Churning Mists - X: 21.0, Y: 25.8)
- Far from Home (The Churning Mists - X: 10.0, Y: 29.2)
- A Nutty Initiation (The Churning Mists - X: 31.6, Y: 24.7)
- Protecting the Pom (The Churning Mists - X: 30.1, Y: 36.3)
- Save the Pomguard (The Churning Mists - X: 20.4, Y: 26.2)
- An Urgent Message (The Churning Mists - X: 8.6, Y: 27.9)
- An Uneasy Feeling (The Churning Mists - X: 9.3, Y: 25.9)
- A Moogle's Intuition (The Churning Mists - X: 7.6, Y: 28.9)
- Trouble at Zenith (The Churning Mists - X: 8.6, Y: 27.9)
This ends the Pomguard quest chain, and unlocks a new quest chain that starts in Ishgard. Other than starting in Ishgard, this is a fairly normal chain without the quest resuming randomly elsewhere in the zone. Locations are given anyway. Flying is required for this chain - you can't reach some of the quest locations without flight.
- Into the Mists (The Pillars - X: 11.4, Y: 11.0)
- Bitter Is the Night (The Churning Mists - X: 22.8, Y: 16.2)
- Cleaning House (The Churning Mists - X: 22.8, Y: 16.2)
- Finders Keepers (The Churning Mists - X: 29.1, Y: 12.9)
- These Things Take Time (The Churning Mists - X: 22.8, Y: 16.2)
- An Unwelcome Surprise (The Churning Mists - X: 7.4, Y: 19.1)
- A Meter Too Far (The Churning Mists - X: 11.4, Y: 9.0)
- Thar Be Dragons (The Churning Mists - X: 22.8, Y: 16.2)
- Laying the First Brick (The Churning Mists - X: 22.8, Y: 16.2)
Once this chain is done, the Moogle beast tribe quests can be unlocked by doing the quest Tricks and Stones (The Churning Mists - X: 27.2, Y: 34.5) as a level 50 or higher crafter.