How do I get my 2012 MacBook Air battery replaced?

How to find your nearest Apple Authorized Service Provider (AASP) or Apple Authorized Distributor (AAD) - https://locate.apple.com/country

Genius Bar appointment - Unfortunately Apple has made this tricky to get to directly. Go to https://www.apple.com/retail/geniusbar/ > Get hardware help. Select your hardware and continue selecting options requiring replacing parts until you see "Bring in for Repair" --> Sign in with Apple ID

Mac Service and Repair - Learn how to get your Mac desktop or Mac notebook fixed and how much it will cost. https://support.apple.com/mac/repair/service

I think that may have just entered Apple's vintage list in which case you will have to see if a third party repair place can help you.

Vintage and obsolete products

https://support.apple.com/HT201624