Yes you can. It is possible to write on the file system of Azure Websites. However your write permissions are limited to the root folder of your app. So, if you use ASP.NET, you shall be able to write anywhere within the Server.MapPath("~/from_here_on"). Meaning you shall be able to perform read/write/delete operations on files which are located in the root folder of your app and below. If you use PHP, the root folder can be get from $_SERVER['DOCUMENT_ROOT'] environment variable.
And a web application shall not need more privileges. For sure will not be able to write on the operating system folders.