@Anonymous have you tried adding the other app's outbound IP addresses to the allowed list? Check here on locating the outbound IP address for your web apps. There are normally four or five outbound IPs that your web app can send traffic out on. The above document also talks about when your outbound IPs might change.
Azure App Services stop communicating with each other after adding white-list addresses to firewall
Geoff Blachstein
31
Reputation points
We have 2 Azure App Services that communicate with each other. They work fine until I configure Access Restrictions under the Networking blade of each web service. Is there a way to white list addresses and still allow the internal web services to talk to each other?
Thank You,
-Geoff