Generally some of the OpenAccess assemblies are loaded from GAC when OpenAccess is installed on the target machine and therefore they are not direcly referenced in your projects.
In case you want to just deploy your application on clean server or as you have experienced with Azure, you should use the methods of deployment described in this article
In general a clean Web API hosting application should require only three OpenAccess assemblies:
The two additional ones that you have included are used if you are configuring OpenAccess runtime through the web.config and using OpenAccessLinqDataSource (or OpenAccessDataSource) in your web application.
We are eager to hear more about your experience deploying OpenAccess in Azure so we can make your and everyone else experience better in future.