Yes, it is possible to write on the file system of an Azure Website. But, your write permissions are always limited to the root folder of your application.
So, if you are using ASP.NET, then you should be able to write anywhere within your
Which means that, you should be able to perform read/write/delete operations on your files which are located in your root folder of your application and below.
In case you are using PHP, then the root folder can be obtained from
$_SERVER['DOCUMENT_ROOT'] environment variable.
And a web application may not need any more of the privileges.
Of course then, you won't be able to write on the operating system folders.