WASHINGTON—Rear Adm. Matthew Klunder has positioned himself to become the Jeff Bezos of the Pentagon.

Much as the Amazon.com Inc. founder envisions mini-drones that deliver small packages across America, Adm. Klunder, the Navy's research chief, wants to create innovative unmanned helicopters able to perform tasks now carried out by humans: resupplying troops in remote areas and rescuing wounded Marines from the battlefield.

His...