Is it possible to add a virtual directory (like in IIS) to an existing website, hosted in Azure App Service. If so can this be done via the Azure Portal?
Virtual Directories are supported for Azure Websites. See Configuring Azure Websites for what you can do through the Azure Management Portal.
From the Azure Portal:
Click on the Website and go to Configure.
Then scroll down to virtual applications and directories (the last config section).
Enter your virtual directory and the physical path relative to the site root and click Save.
As an alternative to Azure Websites, you could use the Cloud Services Web Role or Virtual Machine capability. You could also use Azure Storage to link to your static content.