I have recently moved a .NET Web API 2 app from an Azure Cloud Service to an Azure Web App. However, we have a legacy application that needs to communicate with the API on p
No, you cannot. Only ports 80 and 443 are open.
Check this answer on Server Fault: https://serverfault.com/a/751548/394375.
If you need that control, you can use Cloud Services or Virtual Machines.