Go into the iOS App Store app and tap the purchases option menu at the bottom of the screen.
Then, in the upper left hand corner, use the search box to search your purchased apps and type in Kindle.
Attempt to re-download the Kindle app.
If a pop-up appears asking to download an older, compatible version of the app, then you should be able to recover an older version of your Kindle app.
If no joy, then there is nothing to be done further.
Your 2012, 3rd gen model is now 9-years old and is too old, now.
Sorry.
Apple ended support for all 8-10-year old iOS devices well over 4 years, ago, in 2016.
Third party app developers abandoned and stopped supporting these 8-10-year old iOS devices nearly 3 years, ago, in 2018.
There are, virtually, NO MORE third party apps available from the iOS App Store, any longer!
Third party compatible apps for such old devices are few and far between, now, in 2021.
Simply put, your 3rd gen iPad is too old and obsolete now.
Sorry, but this is the reality for ALL 8-10-year old AND OLDER iOS devices now. Especially going into 2021!
It's 2021 now!
You really need to seriously consider a purchase of a new iPad model OR a much, MUCH “newer” and less expensive, refurbished or used iPad models, from better known, trusted Apple product/device retailers/resellers, locally or online, on the Web/Internet, capable of running the latest iPadOS versions.
There are PLENTY of newer, older iPad models that can run the latest, current versions of iPadOS.
I am sorry that you had to learn all of this, but this does not change any of the facts regarding these very old iOS devices.
Sorry & Best of Luck to You!