Some time passed, and the answer to this seemed to have changed. Although it is still not possible to directly do this with the web.config file, these platform features are now available on the Azure portal.
If you go to: https://portal.azure.com
And then go to:
App Services -> (Your Func App Name) -> Platform Features -> Networking -> Configure IP Restrictions
You can configure IP restrictions here, and this will apply to your functional application.
Similarly, there are now several authentication options:
App Services -> (Your Func App Name) -> Platform Features -> Authentication / Authorization
This does not include basic authentication, but includes a bunch of things.
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