Robinhood Markets Inc. has spent the better part of a decade trying to entice nonprofessional investors to fall in love with markets.

This week, the company will face its first public test of whether those same traders love Robinhood back.

The wildly popular trading platform’s initial public offering is expected to price Wednesday night and begin trading Thursday in one of the most highly anticipated deals of the year. In an unusual move, Robinhood, which boasts 22.5 million users with funded accounts, is expected to sell its users up to 35% of its IPO shares, or up to 18.3 million shares.

The move is part of Robinhood’s self-described mission to democratize investing for all, a philosophy that it reinforced Saturday by hosting a live-streamed roadshow presentation for individual investors. Yet despite having more access than is typical of such trading events, many individual investors are still saying: No thanks.

Across social-media platforms like Reddit, Twitter and Discord lately, talk of Robinhood’s IPO has heated up, with many individual investors encouraging their peers to avoid it. For some, the decision is a revenge play after the brokerage stopped users from buying meme stocks like GameStop Corp. during the height of the frenzy this year.

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. (Published July 15) Photo illustration: Jacob Reynolds/WSJ The Wall Street Journal Interactive Edition

Others say they want to sidestep a company that has been embroiled in regulatory scrutiny and other issues. Some have gone so far as to suggest that they plan in the future to make bearish wagers against the stock.

“I’m avoiding it like the plague,” said Gary Dennis, a 41-year-old Realtor from South Carolina who has briefly used Robinhood but primarily trades from his TD Ameritrade account. Mr. Dennis said he initially planned to invest in Robinhood when he heard about the IPO, but after reading through paperwork filed with securities regulators, he grew concerned about its business model and regulatory inquiries.

Specifically, Mr. Dennis said he is concerned about what he sees as unsustainable forms of revenue. Robinhood raked in roughly $30 million during the first quarter due to trading of the joke cryptocurrency dogecoin, which has fallen more than 70% from its peak this spring. He is also concerned, he said, about the brokerage’s reliance on payment for order flow, the practice that generates most of the company’s revenue. The trading halts that users experienced in January, he said, are troubling, too.

“When they go public, they’re going to be held to a lot more scrutiny if that were to happen again,” he said. “It’s too much of a speculative play for me.”

Robinhood is by no means the first online brokerage to face the public markets. But few others have made their debuts in the midst of a retail-trading frenzy like the one markets have seen this year. Many of the new traders in the industry got their start trading on Robinhood. Now, the brokerage faces the possibility that many individual investors could bite the hand that fed them.

Robinhood declined to comment.

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To be sure, it is impossible to say how much of the negative sentiment on social media will translate into real action. Platforms such as Reddit and Twitter certainly aren’t perfect indicators for how Robinhood’s broad customer base is feeling. It is also possible that individual investors who are initially hesitant might get swept up in the excitement as Robinhood traders get access to IPO shares. Many could later dive in after trading begins to capitalize on what is known as the IPO pop.

Robinhood expects to sell shares at $38 to $42 each, it has said. The company is listing its shares on the Nasdaq, where IPOs in 2020 typically saw a 40% jump from offer price to open price, according to research from Reena Aggarwal, professor of finance at Georgetown University and director of the Georgetown Center for Financial Markets and Policy.

The company has also said it expects to see demand outside of its retail-investor base. According to Robinhood, the venture-capital arm of Salesforce.com Inc. has indicated interest in buying as much as $150 million of its stock at the IPO price.

Many individual investors have also expressed interest in getting in early on the action, including 27-year-old Saleem Baltaji. He said he already requested 20 IPO shares through his Robinhood app. “I’m excited,” the Texas-based trader said. “I think it’s cool that they are giving us the opportunity” to receive IPO shares.

Mr. Baltaji added that he is excited about Robinhood expanding its cryptocurrency business, where he believes the company has a lot of room to grow. “On a long-term basis, I think Robinhood is not going anywhere but up,” he said.

Still, Robinhood also continues to face a number of federal and state inquiries, its filings show, including some related to the trading halts that it instituted on meme stocks this year.

The company disclosed Tuesday that it received an investigative request this week from the Financial Industry Regulatory Authority, Wall Street’s self-regulatory arm. The group is seeking information tied to Robinhood’s compliance with registration requirements for personnel. The company said it was evaluating the matter and intends to cooperate with the investigation.

For traders such as 42-year-old Christian Millet, an operations manager at a construction company in Denver, the January trading halt is one of the reasons he intends to purchase bearish put options against Robinhood when options trading for the stock begins. A Robinhood trader himself for years, Mr. Millet has since switched to the brokerage app Webull. He said he plans to execute his trades against Robinhood there.

“The sentiment across the board is, ‘We’ll stick it to the man and kind of like pay them back...for what they did with the meme stocks,” said Mr. Millet, who noted that he had already switched to Webull and wasn’t affected by the Robinhood restrictions this year. “I’m simply aligning with a cause…I think it’s not a good company to support.”

Write to Caitlin McCabe at caitlin.mccabe@wsj.com