I am unaware of any azure / visual studio tool that would track changes in the wwwroot folder and update a correlating blob storage container (or any other option)
I am not sure if this can be part of a publishing profile configuration, or the task list window.
I am not familiar with powershell so an example would be needed if that is the only solution. If the solution is a script than I would need to know how to tie it into the publish or build process.