There are couple of choices you have.
First, you can have a look at the possibleOutboundIpAddress
of your App Service and whitelist this IPs. This however also opens up the door for IPs not really in use by your App Service.
az webapp show --resource-group <group_name> --name <app_name> --query possibleOutboundIpAddresses --output tsv
Secondly, you can put a NAT Gateway in-front of your App Service. This however requires an App Service Plan that supports virtual network integration.
- Configure regional virtual network integration from within your app service.
- Force all outbound traffic originating from that app to travel through the virtual network. This is done by setting WEBSITE_VNET_ROUTE_ALL=1 property in your web app configuration
- Create a public IP address.
- Add a NAT gateway, attach it to the subnet that contains the app service and make use of the public IP created in step 3.
If you would also like to use a static inbound IP you can find more information here