I guess it runs on some form of Windows Server 2012 but I wanted to know exactly what VMs & Azure Websites run on and what it takes to make Java web apps run on it.
This TechEd session has some great info on Java on Microsoft Azure. Java web applications can be run using a PaaS Worker role or on Azure Websites both of which have some form of Windows Server OS. This fact is supported by these screenshots which show a Windows file path within the metadata shown on a page hosted on Azure Websites and the Windows Server 2008 & 2012 TargetOS options shown within Eclipse configuration options.
You may find this Azure Friday video & this MVA tutorial series helpful to learn more about Java on Azure PaaS