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Because some networks are locked down and only allow whitelisted IP addresses, I hear these questions a lot.
For an individual Azure Web App, you can simply go to the Properties of the application:
You can click the copy button to add them to your clipboard.
If you need to whitelist a region, I first lookup to see what region it is currently deployed in. You can find this information from the Overview page of the application.
Microsoft provide a list of the Azure Data Center IP ranges in XML format
So download this file and search for your region and now you know what IP address to whitelist.