If, on the other, if your web application is a single-page-application, then you might want to consider a framework.
Web API in Umbraco
If you're new to using Web API within Umbraco, I suggest you read, How To Create A Web API In Umbraco In Less Than 5 Minutes!. The gist of that article is that Umbraco provides a special controller you can use that makes exposing a web API in your Umbraco application easy.
If you are used to MVC routing and you're happy with using a standard web API controller, then, by all means, you can add a route in your routes config and use the standard MVC calls. If you use this approach, you should be aware some of the Umbraco contexts might not be set correctly. In summary, the Umbraco controller will probably be a bit quicker to use, but your API URL will need to have Umbraco in it.
The web API code is pretty simple. Inherit from UmbracoApiController and create a method to return data back to your webpage. your methods