Robinhood Markets Inc., the trading app synonymous with hot stocks, got a cold reception from investors in its own stock-market debut.

The investing app tumbled in its highly anticipated trading debut Thursday, closing 8.4% below its initial public offering price. Robinhood stock opened at $38, matching the IPO price, and quickly fell more than 10%. The stock later climbed to approach the IPO price before falling again in the final hour of trading to close at $34.82.

It was a disappointing debut for the company that brought the markets to the masses. Robinhood and its bankers tried to avoid a poor first-day performance by pricing shares at the bottom of its targeted range at a value of about $32 billion. A successful first day was all the more important because Robinhood sold a big chunk of its shares to its own customers.

Breaking with a Wall Street convention of giving individual investors only a minuscule slice of hot IPOs, Robinhood sold between 20% and 25% of its offering to its customers, according to people familiar with the matter.

“One of our company values is ‘participation is power,’” Chief Executive Vlad Tenev said in an interview. “It didn’t seem right for us that IPOs had typically been reserved for the top 1%.”

Robinhood also allowed employees to sell some of their shares immediately, atypical for traditional IPOs. Mr. Tenev and Robinhood Co-Founder Baiju Bhatt each sold 1.25 million shares in the offering, earning them $47.5 million apiece.

That confluence of factors made for volatile trading on Robinhood’s first day as a public company. About 100 million Robinhood shares changed hands on Thursday, nearly double the shares the company and its executives sold in the IPO. Robinhood tried to discourage its customers from unloading shares they received in its IPO right away, warning users that they could be restricted from participating in future public offerings on Robinhood if they sell their shares within the first 30 days.

The risk Robinhood ran in selling so much of its stock to its own customers is that a drop in price could turn them off trading on the app going forward. The company had already alienated part of its user base in January when it restricted trading in hot stocks such as GameStop Corp.

In advance of Thursday’s IPO, many of those disenchanted users chatted on platforms such as Reddit’s WallStreetBets forum about avoiding the stock, while others talked about eventually making bearish trades against it.

After Robinhood’s tumble on Thursday several of those investors celebrated. But some took advantage of the choppy trading to double down on Robinhood.

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Ricky Castillo, a 30-year-old Robinhood user in Texas, said he scooped up 30 additional Robinhood shares around the $34 to $36 range as the stock swung throughout the day. Initially awarded 100 IPO shares, the full amount he requested, he said, Mr. Castillo had planned to sell some and take profits if the stock had jumped significantly. Instead, he saw the drop as an opportunity to lower his average purchase price, allowing for greater profits if the stock rises again in the future.

Mr. Castillo said he had been monitoring the stock after it kicked off trading, but it was once he received a notification—from Robinhood itself—about the stock’s 10% fall that he decided to double down.

“I’ve come now to the point in the markets where I expect nothing but craziness,” said Mr. Castillo, who said he began trading on Robinhood in 2017. “I’m putting my money in the company because I believe in it. I’ve seen them shake up the investment world, and I actually do believe that if they handle everything they need to handle properly, they can be a big-time player in the financial realm.”

Investors that bought into Robinhood this week will own only about 7% of the company. Following the IPO, four venture-capital firms—DST Global, Index Ventures, New Enterprise Associates and Ribbit Capital—hold about 40% of Robinhood’s Class A shares. Some of those firms participated in an emergency funding round for Robinhood earlier this year on terms that allowed them to buy more shares at a 30% discount to the IPO price.

After Thursday, Mr. Tenev held a stake in Robinhood worth about $1.8 billion, while Mr. Bhatt’s stake was worth about $2.7 billion. They hold all of a special class of Robinhood shares that entitles them, together, to about two-thirds of the voting rights in the company.

The brokerage app Robinhood has transformed retail trading. WSJ explains its rise amid a series of legal investigations and regulatory challenges as it looks forward to its IPO. Photo illustration: Jacob Reynolds/WSJ The Wall Street Journal Interactive Edition

There were similarities between Robinhood’s IPO and Facebook Inc.’s 2012 public debut.

Facebook Inc. sold about 25% of its IPO to individual investors. The stock tumbled a day after the trading debut and took more than a year to close above its IPO price again. Facebook closed Thursday at $358.32, more than nine times its $38 IPO price.

“I do ultimately view this as a moment in time,” Jason Warnick, Robinhood’s finance chief, said about its IPO. “We view success over a much longer timeline.”

Write to Peter Rudegeair at Peter.Rudegeair@wsj.com, Corrie Driebusch at corrie.driebusch@wsj.com and Caitlin McCabe at caitlin.mccabe@wsj.com