How can I get a list of TCP connection statuses for an Azure App Service app?
I realize this isn't a direct answer to your question, but we have a similar issue and found our use of HttpClient to be a contributing factor. If you're using a new HttpClient instance for each of your HTTP requests, you could be exhausting your available outbound connections. We found the following article to be very helpful in minimizing the number of outbound connections using HttpClient: https://aspnetmonsters.com/2016/08/2016-08-27-httpclientwrong/
We are also hoping for a way to view TCP connections from Azure App Services, since Kudu shows "Access Denied" when trying to use netstat.
This is now a popular question. To add some extra info, the outcome of the investigation was that one app was using a MySQL client library that had some kind of bug. It was not closing TCP ports correctly. This killed the whole plan. After talking with Azure tech support they agreed that there should be improved reporting of TCP connectivity in the Azure portal. Since then I have not checked if there have been any improvements.