The currently available Kindle App requires iOS/iPadOS14 or later. If you have an older model iPad with an earlier version of iOS installed - and have not previously installed an earlier version of the Kindle App, you’ll be unable to install the current App.
Given that your iPad cannot itself be updated to a iPadOS14/15, if you need the Kindle App, you’ll have no alternative than to replace your iPad with a much newer model.
Whether or not you are able to update any iPad to a newer version of iOS/iPadOS will be dictated by the iPad model - and the currently installed version of iOS. You can find this information here:
Settings > General > About
Not sure which iPad you have? Identify your iPad model.
Here is a list of the highest version of iOS supported by the older models of iPad (for simplicity - “generations” are shortened to a number) that are restricted in their ability to be updated:
iPad1 - iOS 5.1.1
iPad2, iPad3 and iPad mini1 - iOS 9.3.5 (WiFi Only models), iOS 9.3.6 (WiFi & Cellular)
iPad4 - iOS 10.3.3 (WiFi Only), iOS 10.3.4 (WiFi & Cellular)
iPad Air1, iPad mini2 and iPad mini3 - iOS 12.5.5
Newer models, not appearing in this list, are all able to be updated to the current version, iPadOS 15.5