I found a run down of quests you need to complete to get to Moogle Mog.

You got to start from The Price of Principles.

The Price of Principles unlocks Moving On.
Moving On unlocks Flowers For One.
Flowers For One unlocks All Things in Time.
All Things in Time unlocks The Resolute.
The Resolute unlocks Lay the Foundation.

Now, Lay the Foundation unlocks Better Late than Sever, Crate Go Kaboom and Rock-solid Protection (If you run around Revenant's Toll after Lay the Foundation some random NPC's have these, they're classed as main scenario quests, so easy to find.)

After you complete these, you unlock Hest of the Best.
Hest of the Best unlocks Pass the Smell Hest.
Pass the Smell Hest unlocks You're Gonna Cry That.
You're Gonna Carry That unlocks The Things We Do for Tea.
The Things We Do for Tea unlocks It's Possibly a Primal.
It's Possibly a Primal, finally, unlocks Hail to the King, Kupo. (along with The Gifted, which isn't related as far as I know)

Sidenote: I would have removed the name of the quests like the Patch notes do to prevent spoilers but 1. It makes it confusing to know where to go at first and 2. the quest names in no way spoil anything except the Hest quests.

Edit: When you get to the point where you have to: "Speak with <insert name> at the Lotus Stand." during the Hail to the King, Kupo, you might have issue finding it as the person you must speak to doesn't show on the map. You have to go to the Conjurer's Guild and speak to a Silent Conjurer there that takes you to the instance the NPC you need to speak to is in. Kind of confusing if you don't know where Lotus Stand already is.