The Polar Bear Mount was given as a reward for the Moonfire Faire Event in 2021. The Event ran from August 13th to August 26th. At the time of this writing, there is no other way to earn this Mount in the game.
In addition, it’s not purchasable on Mog Station either.
However, it is likely that it will be added to the shop when the next Moonfire Faire Event takes place.
Which is good news if you were hoping to get your hands on this particular bear. The bad news is that it’s unlikely to be added back for free.
But in FFXIV there are a number of other Bears available for you to ride.
So if you have your heart set on an ursine mode of transportation, here are the others:
Warbear – A reward for the Achievement “But Somebody’s Gotta Do It (Warrior)”. Complete 200 high-level duties as Warrior. These include Raids, Trials, and Dungeons at level 50 and above.
Battle Bear – A reward for the “Tank You Warrior III” Achievement. Complete 300 high-level duties as a warrior. These include Raids, Trials, and Dungeons level 61 and above.
Starlight Bear – Initially an Event reward from the Starlight Celebration in 2017 & has since found its way onto Mog Station for purchase. No in-game method exists to obtain it, but it’s yours if you’re willing to open your wallet.
The Starlight Bear in particular is what gives rise to the notion that the Polar Bear isn’t out of reach forever.
It’s likely (just as with previous Event Mounts) that it’ll end up available for purchase at some point.
Unfortunately, you’ll have to bear with the wait until it is. Perhaps work on your Tank Achievements in the meantime!