American cities rushed to provide police departments with body cameras, spurred by public outcry over shooting incidents in Ferguson, Mo., and elsewhere. Having moved fast, however, cities are now running into friction, often from within their own ranks.

Opponents of the contract arrangements say officials may have cut corners by signing no-bid deals, by not testing options thoroughly or by becoming too cozy with vendors. Other cities, after hurrying into camera initiatives, have found unexpected costs, and some are pulling back.

Memphis decided it urgently needed body cameras to provide more police accountability. As with many cities, it turned to a familiar company, Taser International Inc., the stun-gun king, and last year reached a $4.5 million agreement for 2,000 cameras.

Soon, Memphis learned its then-mayor’s campaign manager had business relations with Taser through a consulting firm, drawing public criticism. City officials then concluded that processing the flood of footage would require it to hire far more staff than expected.

Jim Strickland, the new mayor who took office in January, put the Taser body-camera rollout on hold. “I believe in the use of body cameras,” he said in an email. “But in the effort to do something good for our people and our officers, the process was rushed. We want to do this the right way.”

The former mayor, A C Wharton Jr., said he hadn’t known of Taser’s financial dealings with his campaign manager, who said she didn’t do anything improper. Taser, which cut ties with the firm, said it hadn’t known the consultant was a campaign manager.

For U.S. police departments, body cameras have become the tool of choice in responding to demands for stronger oversight of officers. Police typically wear the small video cameras on their uniforms to record arrests, traffic stops and other interactions with the public.

Law-enforcement officials and civil-rights activists alike believe cameras will prompt better relations between police and citizens. The Justice Department has awarded about $20 million to help police buy them.

Taser, of Scottsdale, Ariz., has an early lead in the market, and since the 2014 Ferguson shooting, demand has soared. Of the 50 largest local U.S. police departments, 24 have chosen a body-camera vendor, with 22 picking Taser, according to a Wall Street Journal count. Two cities chose cameras from Vievu, a unit of Safariland Group.

Jim Bueermann, a retired police chief of Redlands, Calif., and president of the Police Foundation, a research organization, said police chiefs are comfortable buying Taser’s cameras because of their experience with its stun guns. Cities are “just trying to save time and money through the process,” he said.

Taser’s camera strategy takes the cellphone-carrier business model. It uses hardware to sell subscriptions to Evidence.com, its service for storing and sharing body-camera footage. Taser said it has signed up roughly 5,000 of the 18,000 U.S. law-enforcement agencies to the service.

“We view Evidence.com as the iTunes of law enforcement,” said Josh Isner, Taser’s executive vice president of global sales.

In the contract’s first year, Memphis agreed to pay $576,000 for 2,000 body cameras and more than $1.6 million to use Evidence.com, according to contracting documents.

Only after the agreement, said Mr. Strickland, Memphis’s mayor, did city officials find they would still need extra staff to pore over hours of footage.

L.A.’s ‘piggybacking’

Unexpected staffing costs also helped halt Taser’s progress toward what would be one of its biggest body-camera contracts, with the Los Angeles Police Department.

Los Angeles was looking into body cameras before Ferguson, when the initiative gathered steam. The police department first purchased 860 Taser cameras with private donations, after testing different brands. A city proposal last year to expand camera use called for a five-year, $31 million deal with Taser that would skip competitive bidding and expedite the deal by “piggybacking” on a contract by California’s Kern County—taking a process already undertaken as the basis for its own.

Los Angeles procurement officials said the city uses piggybacking in “commodities,” or goods, about 10% of the time but said they don’t keep figures on contracts for services like the Taser deal.

Some council members questioned why the city didn’t conduct its own competitive bidding. Kern’s Taser deal was for 30 cameras, while the LAPD intends to buy more than 6,000.

“Why are we piggybacking on the Kern County contract?” an aide to Los Angeles City Council President Herb Wesson, a Democrat, wrote in a December email obtained through a public-records request.

Mr. Wesson and other council members put a hold on the deal late last year over this and other concerns, said Vanessa Rodriguez, a spokeswoman for Mr. Wesson. Chief among them was the total cost of adopting the cameras. On top of the $31 million that would go to Taser, the police department requested $26 million for smartphones, data plans and other technology for the rollout. Police officials said they would need to hire 122 people to handle the video.

The city is going back to the drawing board to reduce costs. “Balancing the need for swiftness in the community” on cameras, said Ms. Rodriguez, “is ensuring that the city is acting appropriately and doing all the due diligence.”

Taser referred inquiries on the contract to the city.

Steve Soboroff, a Los Angeles police commissioner who spearheaded the body-camera effort, said the city was set to get a better deal than Kern County and had negotiated a clause requiring Taser to lower its price if it did elsewhere.

“We did everything legally to do this as expeditiously as possible to not satisfy contract experts but to save lives and create transparency on both sides of the camera,” he said. “This was not done for the critics. This is not the Academy Awards.”

No-bid risk

Mike Purdy, a government-contracting consultant who has worked with Washington state and cities such as Seattle, said no-bid deals expose cities to potentially overpaying and can raise the specter of favoritism and inappropriate influence.

“Agencies that do not bid products, especially highly visible and political products such as police body cameras,” he said, “run unnecessary risks that can cause them problems in the long run.”

Chicago took an approach similar to Los Angeles’s piggybacking. City officials didn’t hold competitive bidding before reaching a five-year contract in January with Taser that city records value at up to $10 million. The contract to buy roughly 2,000 body cameras and up to 800 stun guns came after police shootings that led Mayor Rahm Emanuel to fire city Police Superintendent Garry McCarthy.

Days after Chicago released a video showing a black teenager, Laquan McDonald, being shot dead by police, Mr. Emanuel announced it would expand a pilot program using Taser body cameras. Adam Collins, a spokesman for Mr. Emanuel, said the expansion stemmed from a desire for more transparency around such shootings.

The city based its procurement procedure on a deal New Jersey made with Taser, using an approach Chicago officials call a “reference contract,” which they said they use for a range of products and can save time and money. According to New Jersey records and Taser, the state has a 2012 contract with the company for police supplies. State records show New Jersey has spent about $230,000 on Taser products since then. Taser said the state hasn’t bought any of its body cameras.

Chicago police spokesman Anthony Guglielmi said Taser was the best fit for the department’s existing equipment and that a police team tested other brands, concluding Taser worked best. “Financially it made sense,” he said, “and technologically it made sense.”

In Baltimore, where riots last year followed Freddie Gray’s death in police custody, officials recently chose Taser cameras. The city first tested the technical quality of various cameras without viewing price tags. After Taser scored highest, the city unsealed the company’s price.

Howard Libit, a spokesman for Mayor Stephanie Rawlings-Blake, said Baltimore will now negotiate with Taser over its bid of $12.8 million, which he said was within the range officials anticipated.

“In the grand scheme of procurements we do, it is an uncommon practice,” he said of the sealed pricing and technical testing. “There was a great desire to make sure that the technology was rigorously tested to ensure it was going to work and deliver on what was needed for police and prosecutors and the citizens of Baltimore.”

Taser’s Mr. Isner said even where no-bid contracts were awarded, Taser cameras were often first pitted against competitors’. “We’re happy to compete and we believe the product’s merits prove itself,” he said.

Rankled rivals

Fast-tracked city deals have rankled Taser’s rivals, who say in some cases they could have offered comparable products at lower costs.

Washington, D.C., which approved a no-bid contract with Taser late last year, said it also tested products from Vievu and Wolfcom.

Wolfcom chief executive Tiffany Wang said the capital’s police department didn’t perform training and didn’t install software necessary to upload video from the cameras. “We were very perplexed,” she said. Vievu said Washington police bought equipment and scheduled training for a field test but canceled the sessions.

The police department said an evaluation team “tested the other cameras as well,” but chose not to conduct a widespread test with other vendors because officials concluded Taser had the only adequate solution for annotating video in the field and for cloud-based storage.

New Mexico’s attorney general is investigating former Albuquerque Police Chief Ray Schultz and the city’s Taser body-camera contract. The state auditor found Mr. Schultz took a consulting job with Taser while chief and helped secure a roughly $2 million no-bid city contract with Taser in 2013. The move came after a rash of police shootings in the city led to a call for body cameras.

As part of the investigation, a state grand jury is looking into Mr. Schultz’s role in helping Taser win the Albuquerque contract, according to court subpoenas. Mr. Schultz declined to comment, referring to recent remarks he made in the Albuquerque Journal that there had been “due diligence” to buy the best body-camera system.

A Taser spokesman, Steve Tuttle, said the company has cooperated with authorities. A spokesman for New Mexico Attorney General Hector Balderas wouldn’t confirm or deny the grand-jury probe.

As part of Memphis’s 2015 agreement, Taser signed an $880,000 contract with a marketing firm headed by Deidre Malone, the campaign manager for Mr. Wharton, who was running for re-election as mayor.

Mr. Wharton, who lost to Mr. Strickland, said he didn’t know Taser had hired the firm until after the contract was signed, declining to comment further. Ms. Malone said her consulting deal was based on her firm’s qualifications and nothing else. Taser’s Mr. Isner said: “Obviously, we wished that turned out a little differently.”

Mr. Strickland, a city councilman before being elected mayor, said after taking office this year, he realized “there was no discussion at city council or in the community of the full cost and amount of manpower this was going to need.”

“Obviously, all across the country people are realizing the importance of body cameras,” he said. “But I don’t feel any pressure in Memphis to roll them out before they’re ready to be rolled out.”