I just read an article about Windows Azure Web Site and would like to evaluate this service. My company just developed an ASP.NET MVC based website that enables our customer
If you want to persist files in Windows Azure (Web Sites) you should use Blob Storage. Features that you might like:
Sending emails on the other hand is not possible with Windows Azure. You'll need to do this through a third party. I personally use Amazon SES but SendGrid is also a good option since they have a special offering for Azure users.
I recently was using azure web app to show a client how the development was going. In my scenario, when in production it would execute in a dedicated server with access to the file system so I could not use Azure Blob Storage. The bright side is that Azure Web Apps come along with storage(whose size will depend of the tier the app is in), you can access it through D:\home\site\"your path", that storage is explained in the following link Microsoft. You can access them by FTP (explained here Accessing files via ftp).
Hope it helps :)