Upon your induction into the Immortal Flames, you can obtain your own personal chocobo mount.
Requirements
This quest is available if you chose to join the Immortal Flames and you have completed the quest The Company You Keep (Immortal Flames).
Starting the Quest
Talk to Swift in the Hall of Flames near Ul'dah Aetheryte Plaza to start the quest.
Present a Flame Chocobo Issuance to Mimigun
You can purchase a Flame Chocobo Issuance from the Flame Quartermaster in the Hall of Flames for 200 Flame Seals (in the Materiel tab). You will get 300 Flame Seals for completing the Main Scenario Quest For Coin and Country so finish that quest first, then come back to Ul'dah to purchase your Flame Chocobo Issuance.
After you purchase it, use an aetheryte to go to the Adventurers' Guild aetheryte, then go north and talk to Mimigun. There will be a cutscene.
Name Your Chocobo
Check on your chocobo. You will be asked to name your chocobo. The game refers to your chocobo as he/him and your chocobo is likely to be canonically male, but feel free to ignore this and name your chocobo whatever you want. You can rename your chocobo for free at any time by talking to a Chocobokeep in any city and choosing Chocobo Renaming.
Speak with Mimigun
After you name your chocobo, talk to Mimigun again and complete the quest to receive your Chocobo Whistle. You must use the Chocobo Whistle to unlock your chocobo as a mount.