Please read this short guide before asking all the “how-to”
You need to complete An Animal’s Tale quest before the Gale Key’s chest are unlocked.
1. You need to complete A Thief’s Legacy quest.
2. You need to obtain Earth Key – 8 from Wind Shrine – Exploration.
3. Complete An Animal’s Tale.
4. Return to the Vault at Royal Capital Grandshelt and grab the recommendation letter.
5. The new vault at Industrial City Dilmagia will be unlocked.