Which Apple TV do you own, Identify your Apple TV model? Amazon Prime works perfectly on my Apple TV 4K. If you do not have an Apple TV HD or 4K it does not have an App store, therefore you cannot add and remove apps. The 2nd and 3rd generation ATV's are hard coded. The most you can do is to ensure Apple TV OS is up-to-date and if the problem persists, do a hard restart (unplug from power for 2 minutes) and then plug it back in and test. If it still doesn't work, then you can try to reset it to factory condition per Apple TV Reset or restore your Apple TV to its factory settings and then re set it up and then test.
If it still doesn't work, then it's time for a new Apple TV 4K which is kind of cool because if you buy one it will qualify you for a year of Apple TV + programming.