Hi,
I believe that App Service will continue to support TLS 1.0, based on documentation as well as my experience over the years in regards to how they typically handle this type of scenario. By "scenario" I mean the frontend where TLS terminates is a transparent element of the platform upon which you run your apps. It isn't a service that you write code to interact with like Azure storage, key vault, etc.
The App Service team hasn't made any announcements (other teams have) related to TLS 1.0/1.1 being blocked which is another clear sign. They have telemetry to see how much TLS 1.0/1.1 is being used on the frontends and wouldn't suddenly block that without giving clear communication to affected customers that their apps were about to stop functioning.
Let's say they block TLS 1.0 at some point for App Service. What you can do is run your web api app on VM(s) instead, which will work fine. Currently the major operating systems allow you to enable TLS 1.0 if you need to for legacy applications.
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Thanks.
-TP