Your investigation has led you to believe that someone involved in the kidnappings has been posing as a priest of the Order.
This quest is available if you have completed the quest Amalj'aa Wrong Places.
Starting the Quest
Talk to Isembard in Camp Drybone in Eastern Thanalan to start the quest. There will be a cutscene, after which you will receive a weathered shepherd's tunic and weathered shepherd's slops. If you lose these, you can talk to Isembard to get another set.
Speak with the Locals
Equip the weathered shepherd's tunic and weathered shepherd's slops. You can talk to the locals in any order. For example, you can go south and talk to Knerl, then go south into the inn and talk to Adelstan, then go north out of the inn and talk to Swaenhylt, then go north into the door next to Raulin and talk to Ermegarde, then go east through the other door and go up the curved wooden ramp, then go down the stone stairs at the top and talk to Aurildis.
Speak with Thancred
Talk to Thancred.
Speak With Thancred Again
While still dressed in your weathered shepherd's tunic and weathered shepherd's slops, go southwest to the ramp and go up, then go east and talk to Thancred. There will be a cutscene.
You won't need the weathered shepherd's tunic and weathered shepherd's slops anymore, so feel free to discard them (they are unsellable).
Speak with Minfilia
Use a Vesper Bay Aetheryte Ticket (which you received after completing the quest The Scions of the Seventh Dawn) to return to Vesper Bay.
In Vesper Bay, go into the northwest building and go down the stairs, and check on the Waking Sands Entrance, then go east and check on the Entrance to the Solar. Talk to Minfilia to complete the quest.
Optional Dungeon: Halatali
Now that you have completed "Dressed to Deceive", you can unlock the optional dungeon Halatali. You will need to complete it to rank up in your Grand Company. To unlock it, go outside to Vesper Bay and talk to Nedrick Ironheart to accept the quest "Hallo Halatali".
Next Main Scenario Quest
After you complete the quest, talk to Minfilia to accept the next quest: Life, Materia and Everything.