Lumen without a Database?

And what do you want to do with that data? Will you want to store it somehow?

If you are not going to serve data, you don't need Lumen at all. Build a mobile app or build the front end with that app (either using JQuery or a single page app framework)

I don't understand the question here. If you don't need a database for your Lumen project and just want to store data with a 3rd party, don't use a database. Nothing says you have to use one with Lumen. In fact, eloquent is not even enabled by default when you create a new lumen project.