Mapping Outbound IPs to 1 IP in Azure App Services
Azure Application Gateway allows you to provide one public IP for those App Services.

You have to make the following steps:


Provision Application Gateway
Add listeners with host name for eachof the App Services
Add backends for each of the App Service
Create HTTP Settings with Override with new host name and Pick host name from backend target options
Add rules to connect listeners corresponding to backends


At the end customer has to white-list one public IP assigned to Application Gateway and create DNS entries to point all App Services addresses to one public IP. 

Materials:


Application Gateway support for multi-tenant back ends such as App
service
How an application gateway works