I have an app on my phone that is communication with an API. This API is not fully documented, and some data displayed in the app, I don't know how to reach.
What I wonder is if it's possible in some kind of way to trace what API calls the app is doing?
It's an app for iPhone.
If by "an API" you mean a Web API and thus you want to see which web calls the application makes you can set up a proxy server between the iPhone and the internet and inspect the calls it makes.
See for example Charles Proxy from an iPhone.