Windows Azure Web Sites don't allow you to do this at the moment. You could evaluate other options:
- Use a Virtual Machine with a Data Disk (persisted in Blob Storage, but hard to scale since only 1 instance can mount a specific VHD in Blob Storage)
- Use a Web/Worker Role with the Azure Drive (persisted in Blob Storage, but hard to scale since only 1 instance can mount a specific VHD in Blob Storage)
- Use a Web/Worker Role or VM together with the AzureBlobDrive. This allows you to 'mount' a storage account like a drive. This way your application will think it writes to a disk, while it's actually writing to blob storage.