Is it possible to connect the Azure Mobile Service to an ASP.NET Web API app? I want to use features included in the AMS, but the business logic belongs to the Web API so that I can reuse it in another web app (maybe ASP.NET MVC). Is this possible?
This is definitely possible.
You can use the Azure App Service Mobile Apps service to run your service. This provides the same facilities as Azure Mobile Services, but entirely contained within an ASP.NET Web App (or Node.js web app). It's just an SDK on top of your ASP.NET application.
For more information on this, see the documentation.