I suggest do some googling for "azure blob storage mount as drive on vm image"
you can create a web role vm image that has a mounted drive that is blob storage but works like any hard drive, if you save the vm as an image then when it gets re-created or re-provisioned it will already have the drive mapped on each instance, then you can install the web site as in virtual directory or at least the /Data folder which is the writable portion could be a virtual directory in IIS that uses the blob drive so all nodes of the web role vm will use the same blob storage drive.
The advantage of this approach is that it is just like any hard drive so nothing special has to be done, files can be uploaded the normal way.
If you use Azure Websites (as opposed to setting up your own web role vms), I believe they already work this way, you get scalability but you don't have to worry about doing anything special for file uploads because under the hood Azure Websites uses azure drives that map to blob storage but work just like a normal file system.
Hope that helps,