Azure SQL Server Trigger Not Firing
One of the more likely reasons for this is (as you confirmed) Bulk Insert.  By default, Bulk Insert does not fire triggers.  You can change this behavior in the options for the bulk insert when your application/code is setting up the operation.  

How to do that can vary by the specifics of your code, Powershell, .Net, etc.  But this other answer may be able to help.  https://stackoverflow.com/questions/12366510/trigger-not-firing-for-multirow-bulk-insert