Nobody can say Rodney McMullen doesn’t know the grocery business. He started as a bagger at Kroger Co. when he was in college, rising through the ranks to become chief financial officer, then chief executive officer, of America’s biggest supermarket chain.

But can he lead the 2,764-store Cincinnati-based company through the changes upending the supermarket industry?

“You are in Cincinnati. You are a conservative bunch of people,” said Bill Smead, chief executive of Smead Capital Management and a Kroger investor. “Does anyone’s blood pulse through their veins with an entrepreneurial bent?”

Not since Walmart Inc. first pushed into groceries in the late 1980s have traditional chains faced so many challenges. E-commerce is transforming the business, forcing cash-strapped companies to overhaul their operations and invest heavily in technology and talent to keep customers from straying to Amazon.com Inc. At the same time, they have to keep food prices as low as consumers have come to expect.

The transition has proven rough for Kroger, which stayed focused on store sales long after mass-merchant competitors were investing in online-ordering technology and delivery services. Executives have debated which investments to make and how drastically to change the company’s business model. Some would-be technology partners have been turned off by what they see as the grocer’s conservative culture—including members of one group who stormed out of a meeting in protest.