C# Azure Function trigger when SQL Database has a new row added without polling

Function Apps don't have an SQL trigger, but Logic Apps do, and it works for both on-premises SQL Server and Azure SQL Server. This will trigger when new rows are added, but do note it uses timer-based polling.

It

It is possible but it's experimental right now. There's a guide that I'm putting at the end of this answer and you'll see that it's pretty straight forward. But again, this is experimental and don't expect that it works well all the time.

Another option will be in the code you have that insert a record to also send a message to queue o service bus and you can then make use of that as a trigger to your function (with service bus you also configure a dead letter queue for retries).

Reference: https://docs.microsoft.com/en-us/azure/azure-functions/functions-bindings-external-table