Nordstrom Inc. invested heavily in e-commerce, didn’t open too many cavernous stores and has been quick to experiment with new types of shopping formats, including stores that don’t carry any clothes.

Yet, it’s suffering the same fate as department stores that innovated less, with year-over-year sales and profits expected to fall for the second consecutive quarter when it reports results on Wednesday. Its stock is the second-worst performing in the S&P 500 retail index behind shares of Macy’s Inc., falling 52% over the past 12 months through Friday’s close. During that period, the index was roughly flat.

Its performance has led some directors to push for bringing in an outsider to replace the two great-grandsons of the founder who run the company, as The Wall Street Journal previously reported. Unwilling to relinquish stewardship of the 118-year-old retailer, the family is considering a proposal that would boost its roughly one-third stake to over 50%, though it isn’t yet clear if the board will be receptive.

Also unclear is whether a fresh set of eyes would solve Nordstrom’s woes. Some of its problems stem from the fact that it operates department stores in malls where fewer people are shopping as they buy more online and from upstart brands. Other department stores, including Macy’s and J.C. Penney Co. , are suffering too.

Nordstrom has had only one nonfamily CEO since it was founded in 1901. John Whitacre spent two decades working his way up the company ranks before he was named CEO in 1997. He resigned in 2000 following poor results.

“John had the clear support of the family and yet it was still a challenging dynamic,” said Kathy Gersch, who was Nordstrom’s vice president of finance during that time.

Nordstrom doesn’t currently have a CEO. It is run jointly by Pete and Erik Nordstrom, who share the title of co-president. A third brother, Blake, who was also a co-president, died unexpectedly in January, after disclosing he had been diagnosed with cancer.

Pete focuses on merchandising and the off-price Rack division. Erik concentrates on the full priced stores and e-commerce. They divvy up corporate functions such as human resources, marketing and technology. Blake had handled more of the traditional CEO duties such as communicating with Wall Street.

The brothers make all major decisions together, often hashing out issues in a Monday morning meeting so that they present a unified front when they sit down with the executive team later that day, said former and current employees.

Such management by consensus may not be ideal in a world that is changing rapidly, but it’s an improvement over the six Nordstrom family members who held the co-president title in the late 1990s, a setup that Ms. Gersch, who is now an industry consultant at Kotter, called “insane.”

The family’s enduring presence created the cult of customer service that endeared Nordstrom to generations of shoppers, former employees said. But it also bred insularity. In meetings internally and with suppliers, executives can be polite to a fault, unwilling to make tough decisions or push for a better deal, the people said.

“There is a saying inside the company, ‘Nordstrom Nice,’” Pete Nordstrom said in a 2017 interview. “Sometimes I worry that we’re too nice. We have to be super accommodating with customers, but we also have to pivot and be sharp businesspeople who negotiate in the best interest of our business.”