Scientologists are emailing advertisers and demanding they boycott the A&E show “Leah Remini: Scientology and the Aftermath,” claiming the documentary series is inciting threats and acts of violence against members of the church.

Individuals who say they are Scientologists sent multiple versions of the letter in recent months to advertisers and ad buyers, according to people familiar with the matter. The group behind the effort, Scientologists Taking Action Against Discrimination (STAND), also posted a handful of letters addressed to Anheuser-Busch InBev SA, Fiat Chrysler Automobiles NV’s Chrysler brand and Berkshire Hathaway Inc.’s Geico, among others, on its website.

In “Scientology and the Aftermath,” now in its second season, Leah Remini, an actress known for her role in “The King of Queens” and an ex-Scientologist, and other former Church of Scientology leaders and members interview ex-members and their families. On its website, A&E describes it as a show through which “Leah gives the victims a chance to be heard.”

“We are enormously proud of the quality and importance of the series, which was affirmed by the show’s recent Emmy win, and we intend to continue to share these brave stories with viewers,” said an A&E spokesman in an emailed statement. The program won an Emmy this year for outstanding informational series.

A&E declined to make Ms. Remini available for comment.

“Scientology and the Aftermath” made its debut in November 2016 to the strongest ratings on the channel in two years. Through September, Ms. Remini’s show has averaged 2.8 million total viewers per episode, including live viewing and on-demand episodes watched within seven days, and 1.6 million adults aged 25 to 54, according to the network.

The channel’s parent company, A+E Networks, a joint venture of Hearst Corp. and the Walt Disney Co., has been no stranger to edgy programming, including putting “Duck Dynasty” on the air. But such risks haven’t always paid off. A documentary series about people trying to escape the Ku Klux Klan drew backlash and was axed before it ever aired when the network said it learned the production company had paid some participants.

A person at an ad-buying firm said that some clients pulled ads from “Scientology and the Aftermath” because of the public controversy surrounding it—not necessarily as a result of the letters. “Sometimes any news around a particular show puts it under new light, and why take the risk?” this person said. Another said that the letters wouldn’t affect the company’s ad spending with the A&E show.

A few advertisers pulled out of the show but continued to advertise elsewhere on the network, according to an A&E spokesman.

Geico said its ads likely appeared on the Scientology show as it regularly rotated its commercials through several A&E programs. “That rotation ended previously and Geico ads are not appearing in time slots for that program,” said a Geico spokeswoman.

Geico didn’t respond to questions about whether the show’s content or the letters prompted its decision not to advertise on the show.

Chrysler and Anheuser-Busch didn’t respond to requests for comment.

“Leah Remini’s hate campaign of religious bigotry in its first season alone generated more than 400 incidents of harassment, threats of violence and vandalism against our churches and members,” reads one letter from STAND, dated from August and addressed to Geico’s assistant vice president of marketing Bill Brower. “The threat level has again risen, precisely coincident with A&E’s promotion and airing of the second season of this show, now spawning even more threats—bombings, murder and acts of physical violence.”

A spokeswoman for the Church of Scientology said in a statement that, while it hasn’t seen every letter from Scientologists to marketers, “advertisers are being written to educate them and expose the lies that A&E irresponsibly airs from a hateful Remini and her unvetted—albeit compensated—subjects.”

“Scientologists, like members of any religion or group being discriminated against, have every right to communicate their disgust at lies being spread about their religion,” she said. The show has triggered “an explosion in hate crimes, threats and even violence directed at Scientologists,” she added.

Groups of activists have historically urged brands to pull ad spending from content that they might find offensive or not in keeping with their beliefs. Some of these campaigns can succeed, especially with the advent of social platforms such as Twitter and Facebook, which make it easy to spread a message quickly and publicly pressure brands and individuals to take action.

Ms. Remini, who was raised in the Church of Scientology, has been critical of it on her show and in interviews she has given for news programs.

Her series is far from the first to focus on Scientology and attempt to shed light on the religion.

“The Master,” a 2012 film described as being loosely based on the founding of Scientology, as well as the HBO documentary “Going Clear,” were among many works in recent years that cast the church in a negative light.

Write to Alexandra Bruell at alexandra.bruell@wsj.com