At the very first launch of the app, yes. Hangar 360 uses the DJI SDK to connect to the aircraft. Which requires an internet connected registration to DJI at least once when starting the app.
After that connection, you may freely take photos without an internet connection.
Keep in mind that to receive a full stitched and processed photo from Hangar, and not just individual images on an SD card, you’ll need to upload the photos to Hangar servers using an Internet connection at that time.