By default the JavaScript API uses a lot of resources from ArcGIS Online (AGOL), which requires an internet connection.
It's possible to host these resources yourself and totally eliminate any dependence on AGOL. In my experience this is most often done via Portal + ArcGIS Server (Portal is essentially your own version of AGOL) but it should be doable to set this up just with server.
Anyway, the approach I would take is to use your browser's dev tools to see which requests your browser is trying to make. The most likely candidates are requests for basemaps: by default, basemaps are pulled in from AGOL. You'll have to host your own basemaps. Portal provides a streamlined way to do this: you can tell the JS API to pull basemaps from Portal rather than AGOL (note: you'll still have to ensure that your Portal + Server installation is hosting some basemaps, a default Portal install still points to AGOL for basemaps).
A lot of other resources in the JS API point to AGOL by default as well, such as geocoding and the world elevation layers.
