I have a WebRole running in Windows Azure, using a *.cloudapp.net domain that I have CNAMEd my real domain to. I don't have a Virtual Machine.
Is it possible for me to issue a certificate signing request to a CA with this setup? Or do I need to purchase a Virtual Machine from Azure to do this?
You can create a certificate signing request from any IIS server, including your development machine. All you need to do is install the certificate that you get from the CA on the same machine, so that you can export the .pfx for Windows Azure. You don't need an Azure VM or use Azure at all to create a certificate signing request.