COLLEGE STATION, Texas—When U.S. Army Chief of Staff Gen. Mark Milley started complaining that it shouldn’t take the Army 10 years to buy a new pistol, it served as a wake-up call to rewire a plodding and bureaucratic process for acquiring new weaponry and modernizing the country’s largest military branch.

Gen. Milley and Acting Army Secretary Ryan McCarthy are attempting a major restructuring of the Army procurement system—potentially the biggest in decades, they say—by eliminating layers of bureaucracy and bringing the...