By default not all HTTP methods or verbs are allowed in Azure App Service. In order to allow/remove methods/verbs we need to make configurational changes.

It really boils down to what stack is your App running on.

Web Apps (Windows)

In IIS handlers are responsible for different app stacks and this is where the allowed verbs are also configured. We can see how the different handlers are configured in the applicationHost.config file. More details on it in my other article.

Go to KUDU and open the applicationHost.config file - D:\Local\Config\applicationhost.config. Find <handlers accessPolicy="Read, Script"> row. From there you can examine how different technologies are configured.

Let's take PHP for example

We know that we have a couple of versions of php available in App Service.

This is PHP 5.6

<add name="PHP56_via_FastCGI" path="*.php" verb="GET,HEAD,POST" modules="FastCgiModule" scriptProcessor="D:\Program Files (x86)\PHP\v5.6\php-cgi.exe" resourceType="Either" />

This is PHP 7.3

<add name="PHP73x86_via_FastCGI" path="*.php" verb="GET,HEAD,POST" modules="FastCgiModule" scriptProcessor="D:\Program Files (x86)\PHP\v7.3\php-cgi.exe" resourceType="Either" />

Let's say we use 7.3 and we want to allow DELETE method.

Let's put this into our web.config file under <configuration> and <system.webServer>

<handlers>
  <remove name="PHP73x86_via_FastCGI" /> <!-- We instruct to delete the existing handler with that name -->
  <add name="PHP73x86_via_FastCGI" path="*.php" verb="GET,HEAD,POST,DELETE" modules="FastCgiModule" scriptProcessor="D:\Program Files (x86)\PHP\v7.3\php-cgi.exe" resourceType="Either" /> <!-- We give the new values for the handler. It is basically the  old handler but we added the DELETE method to the verbs -->
</handlers>

If we want to do that for another stack/technology - we want to notice the important parts of the handler

<add name="PHP73x86_via_FastCGI" path="*.php" verb="GET,HEAD,POST" modules="FastCgiModule" scriptProcessor="D:\Program Files (x86)\PHP\v7.3\php-cgi.exe" resourceType="Either" />

The ones in yellow are the ones we want to pay attention to. My suggestion is to go to the applicaitonhost.config file, find your desired handler and copy the hanlder from there. Use the example above but modify for your handler of choice values.