By Jacob Bunge

May 6, 2015 7:25 p.m. ET

The proxy battle between DuPont Co.  and Trian Fund Management LP pits the industrial giant against one of Wall Street’s most successful activist investors, and it involves financial institutions that manage billions of dollars. But the outcome could come down to such individual investors as retired electrical-utility manager Roger Rinehart.

The 68-year-old resident of Dayton, Ohio, has about 100 DuPont shares in his personal portfolio, and he has been inundated with pitches from both sides—so much so that he asked DuPont to stop calling him after about their third try. He said he hasn’t yet voted his shares for either side’s board nominees, though he’s leaning toward Trian’s. “I have no quibble with the stock price,” he said, but the company’s performance “seems stagnant.”

Individual investors like Mr. Rinehart are typically wallflowers in big corporate-governance debates. But at DuPont they account for a third of total shares,�more than is usual at big companies. And with the May 13 vote to pick directors expected to be close, they have become a wildcard in one of the biggest proxy contests in history.

Trian has nominated four candidates for DuPont’s 12-member board, including Trian Chief Executive Nelson Peltz. It wants DuPont to slash expenses and consider breaking up the company. DuPont responds that a breakup is a bad idea that needlessly distracts from its plan to refocus on more-profitable products, which executives say is succeeding. Neither side has shown signs of backing down, though a settlement is always possible.

Shareholders can vote using a proxy card from one side or the other. Trian’s card counts as votes for the eight DuPont directors Trian is supporting, in addition to Trian’s four nominees, unless a shareholder explicitly withholds votes from any of the names. The new board will be composed of the 12 nominees who win the most votes.

While both sides are traveling the country to court professional money managers, they have also carpet-bombed mom-and-pop shareholders with emails, phone calls, and advertisements in the News Journal, the hometown newspaper in Wilmington, Del., where DuPont is based. A Google search for “DuPont Trian” brings up jousting websites created for the proxy contest: dupontdelivers.com belongs to management, and dupontcanbegreat.com is Trian’s.

“As a DuPont shareholder, you have a powerful voice in this matter,” DuPont Chief Executive Ellen Kullman said in a letter Tuesday, one of seven the company has sent to shareholders. “Your Board and management are determined to prevent the negative and far-reaching effects on our shareholders, customers, employees and the communities in which we operate,” the letter said of Trian’s campaign.

“It is time for change at DuPont,” Trian executives countered Wednesday in the latest of their five letters. “Your vote is important, no matter how many or how few shares you own.”

Robert Sacino, a retired professor, said he’s pleased with DuPont, shares of which have nearly tripled during Ms. Kullman’s tenure. “I’m not opposed to activism,” said Mr. Sacino, 63 years old, who lives in New Rochelle, N.Y. “But I don’t think the target is right for Trian.”

Greg Poore believes Trian’s Mr. Peltz will help tighten DuPont’s focus on meaningful innovation and streamline the 212-year-old company’s operations. “DuPont’s one of those classic corporate cases of the ‘ivory tower’ corporate directors who have lost touch,” said Mr. Poore, a Fountain Hills, Ariz., based investor who voted his shares for Trian’s nominees.

Parties in proxy fights often reach out to individual investors, but they tend not to focus on them because such investors rarely turn out in big numbers, whereas mutual-fund firms have fiduciary duties to vote their shares, say corporate governance experts.

DuPont, with a market capitalization of about $66 billion, has estimated it has about 600,000 individual investors, holding about 33% of its equity.

That compares with about 28% at typical large-cap companies, according to Broadridge Financial Solutions Inc., which manages companies’ communications with investors.

U.S. retail shareholders last year voted about 29% of their shares in corporate governance matters, down from about 40% to 45% roughly a decade earlier, according to Broadridge.

The boardroom battle has been costly. Trian estimates it will spend $8 million on it. DuPont puts its tally at $15.4 million. DuPont’s proxy solicitor, which canvasses shareholders to gather votes, has about 200 employees working on the fight, while Trian’s has about 175, according to regulatory filings. Other recent proxy campaigns typically had closer to 50 solicitors working on each side.

Some mutual-fund managers back DuPont’s management while others say that Mr. Peltz’s track record of helping to revamp companies should earn him a DuPont seat. Since late April, three proxy advisory firms, which advise financial institutions on corporate governance matters, have endorsed Mr. Peltz’s candidacy. A fourth has backed all of DuPont’s nominees.

Many fund managers have to own shares of companies like DuPont that are part of key stock indexes. But individual investors who are dissatisfied with a firm’s performance can just sell, so those that do hold shares tend to view sitting executive teams somewhat more favorably, according to academics.

“When they do vote, retail investors tend to side with company management more than institutions,” said Alicia Davis, a law professor at the University of Michigan who studies shareholder behavior. “This makes individual investors key players in close votes accompanying battles for corporate control.”

Some shareholders say their holdings are about more than share performance and dividends. An estimated 2% of DuPont shares are owned by investors based in or around Wilmington, including many current and former DuPont employees or their relatives.

Alan Behringer, a Wilmington-based lawyer, inherited DuPont shares from his father, who worked for the company for 41 years. Mr. Behringer says he fears Trian could drastically alter DuPont’s research and development, potentially hamstringing long-term competitiveness. He acknowledges a “built-in bias in favor of existing management.”

“I owe a lot to the fact that the DuPont company was able to provide [my father] with a good job for 40 years,” said Mr. Behringer, 69, who voted his 180 shares for DuPont’s directors. “That’s the way a lot of us in Delaware feel.”

—David Benoit contributed to this article.

Write to Jacob Bunge at jacob.bunge@wsj.com