I’ll answer my own question, which was sorted in collaboration with a Shopify Partner and no help what so ever from Shopify team. (A bit disappointed about the lack of support tbh).
My problem was, that I got empty respons on …/orders//fulfillment_orders.json.
After testing on to new shops, where everything worked fine, we tried creating a new private app. Then everything worked as expected. So if you are struggling with empty array response on that endpoint, try updating permissions or creating a new private app.