Nope we do not support Azure yet.
Frankly, I have no idea how it is done, maybe you can use External tools feature to trigger a command-line ulitily to publish stuff? Or maybe you can upload your files via FTP/SFTP/FTPS to the server? WebStorm supports this as well.
I can see that these various modes are supported. I am not sure how the credentials would get uploaded. The same situation occurs for any cloud server Azure, Amazon, Google. etc. All need some form of credentials.
So does Azure use FTP to publish?
There must be more involved than this. As a bare miniumum I have a public private key pair that form my credentials. I also have a user name and password that I have to enter. FTP only offers user name and password.
Actually the most convenient way to publish to Azure is Git. Please check this blog post by our technical evangelist Maarten Balliauw: http://blog.maartenballiauw.be/post/2012/12/28/Windows-Azure-Websites-and-PHPStorm.aspx - it's pretty simple to deploy to the Azure instance right from the IDE.
I tried to follow along but got to one of the first steps and I clicked on 'Test' in the Clone Repository button and it never returned. Constant mesage box with a bouncing progress bar. If I interrupt this process by clicking on 'Cancel' then I get a red failure button and the 'Clone' button is disabled. I was using the 'GIT CLONE URL' from the Management Portal.