How do you create an entry in the registry (HKEY_LOCAL_MACHINE) of an Azure Website? The web application only needs to read the entry.
The following page states the common baseline OS functionality for all apps running on Azure App Service and includes a section on "Registry access": https://azure.microsoft.com/en-gb/documentation/articles/web-sites-available-operating-system-functionality/
Apps have read-only access to much of the registry of the host VM. However, no access whatsoever to the HKCU hive is supported and write-access to the registry is blocked (as at 26/01/2016).