Boeing Co. is poised to lose its place as the world’s largest plane maker to Airbus SE after a reign of seven years, as its jetliner deliveries fell by more than a third in the first half of 2019 with the grounding of its 737 MAX aircraft.

The U.S. aerospace giant’s best-selling MAX has been barred by safety regulators from flying passengers for almost four months, far longer than Boeing, its investors and airlines expected. Consumers have expressed misgivings about getting on the plane again, and a dearth of new sales to...