By Sarah Krouse, David Benoit and Tom McGinty

Oct. 24, 2016 10:41 a.m. ET

Investor Jeffrey Osher and his advisers arrived at the May annual meeting of Green Dot Corp., a prepaid-card company, thinking they had enough votes to remove its chief executive from the board.

In the parking lot, they discovered that was no longer true. Index-fund giant Vanguard Group, the fourth-biggest shareholder, changed its vote over the prior weekend to support the CEO.

The reversal is a reminder of who now has leverage over America’s corporate boards. It increasingly belongs to investors such as Vanguard, pioneers of passive investment funds that track indexes instead of trying to beat the market like Wall Street’s classic stock pickers.

Passive funds are increasingly the default investing option for individuals and big pension funds alike. At the end of June, U.S.-based mutual funds and exchange-traded funds that track indexes owned 11.6% of the S&P 500, up from 4.6% a decade ago, according to a Wall Street Journal analysis of data from Morningstar Inc. and S&P Global Market Intelligence.

Vanguard’s U.S.-based passive funds owned 5% or more of only three S&P 500 companies at the end of 2005. By the end of June this year, that number had rocketed to 468 companies, or about 94%, according to the Journal analysis.

Passive funds used to be unequipped and unwilling to weigh in on most corporate events. With their new power, they are waking up to how much sway they can exercise over takeovers, the fates of chief executives and other crucial decisions.

In June, BlackRock Inc., another large passive investor, voted against the executive pay plan at Mylan NV, which has since been embroiled in controversy over its pricing of the EpiPen allergic-reaction drug. BlackRock also cast an influential vote last year in favor of the $18 billion merger between professional-service providers Towers Watson & Co. and Willis Group Holdings PLC.

Index-fund executives have added employees to deal with shareholder votes, but say they don’t use their new power like activists investors, who accumulate shares and make demands. They typically view themselves as long-term holders that shouldn’t meddle in day-to-day management.

“It’s not shareholders’ role to second guess what management is doing in every single issue,” said Michelle Edkins, who supervises how BlackRock, the world’s biggest money manager, votes shares held in its passive funds.

The change is shifting the clout in boardrooms toward investors known for being deferential to management, a stance troubling to stock pickers who believe shareholders should assert themselves.

“As more and more capital flows to index funds, the seriousness with which these funds approach governance issues becomes even more critical for U.S. and global corporate competitiveness,” activist William Ackman wrote to his hedge-fund investors earlier this year.

Daniel O’Keefe of investment firm Artisan Partners Ltd., an active manager, contends that “the tyranny of passivity is you have large pools of money that are unengaged in their investments,” which he argues is “a far greater risk than the tyranny of activism.”

The last time such a concentrated group of owners had as much control over U.S. companies as index funds do now was during the era of J.P. Morgan and John D. Rockefeller, when those two businessmen had board positions at a number of companies in which they were large owners, researchers at the University of Amsterdam wrote in a recent paper. They eventually relinquished some of their power because of government and public pressure.

In the ensuing decades, stock ownership spread among individual investors, leaving corporate managers largely free of powerful stockholders with the ability to reshape their companies, the paper said. It wasn’t until the rise of mutual funds in the 1990s that single entities amassed large positions across so many companies.

When Vanguard started indexing in the late 1970s, founder John Bogle gave the job of overseeing shareholder votes to one person, a staff assistant. For years, Vanguard index funds voted some of their shares, but didn’t devote significant resources to meeting with the management of companies they held.

That has now changed. “We moved from a position of reluctance to make our weight felt to absolute interest in making our weight felt,” said Mr. Bogle, now retired from the firm.

Vanguard has 15 people overseeing work on about 13,000 companies based around the world. BlackRock has about two dozen people who work on governance issues at some 14,000 companies held in its index funds and exchange-traded funds, and it plans to add seven more in the coming months, according to a spokesman.

Boston-based State Street Global Advisors, another large passive-fund manager, part of State Street Corp., has fewer than 10 employees devoted to issues at around 9,000 companies and uses a number of automated filters to identify companies on which to focus each year.

Mr. Ackman, the activist investor, visited Vanguard in the summer of 2015. It was several weeks after Vanguard, BlackRock and State Street had backed DuPont Co. management in a shareholder vote, helping the chemical company defeat a campaign by activist investor Nelson Peltz and the firm he co-founded, Trian Fund Management LP, to secure board seats. That vote spurred concern among several activists that passive funds were simply too passive.

Mr. Ackman and Vanguard board members and executives presented different views on what type of shareholder behavior leads to corporate change.

Mr. Ackman’s Pershing Square Capital Management LP has an investment team of eight, plus Mr. Ackman and various other employees, for a portfolio that normally includes about a dozen stocks.

Glenn Booraem, a principal at Vanguard who works on its governance efforts, said in an interview that his team can’t effectively meet with all the companies in Vanguard’s funds in one year. He said he tries to take a longer view, communicating regularly with portfolio companies in person, over the phone or via email.

Rakhi Kumar, head of corporate governance at State Street’s asset-management unit, said she tells her team not to agree to every meeting companies ask for because of time constraints. “If I don’t have something to say to you, it’s meaningless,” she said.

Index funds say they do watch over management. Their executives say they have different investment time horizons and fee structures than activists, which affects their decisions about portfolio companies.

Before the vote at prepaid-card company Green Dot, Vanguard talked with board members after the activist investor, Mr. Osher, argued the company’s CEO and founder, Steven Streit, had to go, citing stock declines and its disappointing earnings.