There were rumors that it would be available last month, but alas, no. Latest I have seen is by end of year.
Given A'zon is now (announced yesterday) looking to add a free add-supported video plan, I suspect the delay may be tied to a late change of direction in the project.
Amazon balked because they didn't want to pay Apple 30% for every $99 Prime membership if memberships were created in-app. If they only allow in-app rental\purchase of video products, they only have to pay Apple 30% of these add-on sales, but leave the $99 subscription available online only. Or, they could do like iOS where they only allow online rental\purchase and you have to bounce back and forth (not optimal for users, and likely rejected by Apple).
More likely, Apple agreed to a better margin for A'zon to sell ATV in exchange for allowing A'zon to rent\purchase in-app ala-carte content only, and maybe at a better margin than the standard 30%.
With the new free-tier, Amazon may want to allow users to create a "free" Prime account in app which would have been out of scope before the announcement. If this is the case, the ATV app would have a method to either sign in (to an existing Prime Account) or create a Prime Account. The create option would be for the free Prime Video tier only, and if you want to upgrade to add-free, go online and fork out the $99 annual fee. All of this would be significant changes to the approach they were probably taking before this idea was conceived.