You cannot add a Windows Azure Web Site to a virtual network comprised of other Windows Azure deployments.
As long as your database resides within the same datacenter, a virtual network isn't needed. If you create a Windows Azure Web Site, you can simply talk to a Windows Azure SQL Database - just set one up through the portal, get the connection string and credentials, and add it to your app. In fact, when you create a Web Site from the gallery (that uses SQL Server vs MySQL) you're asked to either create a new database or select an existing one.
In the portal, you can easily create your own database (this is database-as-a-service, and takes only a few seconds). Once you do, you can enable the firewall rule to allow Windows Azure services to access it. Then don't add any further firewall rules. At this point, only Windows Azure services could even find your database, and only services knowing the credentials can connect to it.
As @user728584 suggested, you could also run a database within a Virtual Machine, but you'd then have to access it via public IP address. If you access a database on-premises, you'd have the same issue, as there's no way to install a vpn client into an Azure Web Site. You can do this with a web application hosted in a Web Role (you can do pretty much anything).