Amazon Prime Video for Android TV won't be widely available anytime soon1
It looks like Google will not make the Amazon Prime Video app widely available through the Play Store, 9to5google reports. The client will be pushed out on an OEM by OEM basis, which means that it's all up to your Android TV manufacturer to make the Amazon Prime Video app available for download.
Hisense is one of the first TV manufacturers that has already deployed the new Amazon Prime Video app to some of its Android TVs, as pointed out by Reddit user CenterInYou. The update that contains the Prime Video app also adds a new boot-up screen and June security patch in case you want to know.
The Prime Video app will only be made available through Google Play Store if and when OEMs decided to push it out, so there's not much you can do other than wait and hope that your Android TV manufacturer will roll out the app sooner rather than later.