Companies are encouraging their workers to get vaccinated as Covid-19 cases climb again. Yet relatively few workplaces are making shots required.

On Monday, the U.S. Department of Veterans Affairs became the first federal agency to require employee vaccinations, and California became the first state to do so for its public workers. In New York City, Mayor Bill de Blasio said that all municipal employees will be required to be vaccinated by Sept. 13, or get tested weekly, and urged the city’s private employers to push similar measures.

President Biden said Tuesday that a vaccine mandate for all federal employees was under consideration.

Some companies, including Morgan Stanley, Goldman Sachs and Wells Fargo, have mandated that all workers must get vaccinated or divulge their vaccination status before returning to the office. United Airlines requires Covid-19 vaccines for new employees.

Overall, though, Covid-19 vaccine mandates at work remain rare, partly because doing so can be a legal minefield for companies, employment attorneys say. In some cases, mandates have resulted in employee lawsuits and termination of workers. And in a tight labor market, employers also risk losing workers who balk at such requirements.

What appears to be emerging for many workers is a patchwork approach to vaccine mandates based on job type, geography, work setting and clientele. Some meat-processing companies required the vaccine for new corporate hires this spring, but stopped short of mandating vaccines for the whole workforce. Other companies with employees in tight spaces did require the shots.

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Newell Brands Inc., maker of Sharpie markers, Mr. Coffee appliances and Graco baby strollers, recently started allowing workers who have proof of vaccination to go without masks inside the company’s manufacturing facilities. The Centers for Disease Control and Prevention recommended Tuesday that vaccinated people resume wearing masks in some indoor spaces, reversing its guidance from May that most vaccinated people no longer needed to mask or physically distance in most indoor and outdoor settings.

In hospitals and healthcare settings, where more mandates have been rolled out than in other sectors, fully vaccinating staff is about protecting clients who are immunocompromised in addition to employees, executives say.

Heather Berg, a group leader at a nonprofit that runs after-school programs in California, says she wants more employers to require vaccines, especially those operating in healthcare or with children. “I feel like if you’re not interested in protecting people’s health why do you work in healthcare?” she says. “And when it comes to kids, they don’t have a chance. So if you can’t choose to protect kids who can’t protect themselves, what are you doing in education?”

Ms. Berg says her employer, who she joined last month, doesn’t mandate vaccination, which made her hesitant to take the job at first. But she says she felt reassured when the people who hired her volunteered that they had been vaccinated. For now, she and other staff members and the children wear masks indoors.

“I hate it, I wear glasses and they fog up,” she says. “I wish everyone would get vaccinated so we can be done with it.”

A recent survey by the American Medical Association found that 96% of physicians have been fully vaccinated against Covid-19, but among healthcare workers overall, a recent analysis of U.S. Department of Health and Human Services data found that one in four hospital workers with direct contact with patients hadn’t received a dose of vaccine by the end of May.

As companies plan to reopen offices to workers in larger numbers come September, some are telling employees to get vaccinated before they return. To do so, some are offering perks like an extra day off for showing their managers that they have proof of vaccination, human-resources experts say.

While Covid-19 vaccinations are highly effective at preventing hospitalizations or death from the virus, they’re not foolproof in preventing infection. This poses problems for events like the Olympics and raises broader questions about immunity in the long term. Photo: David Crigger/Bristol Herald Courier/Associated Press The Wall Street Journal Interactive Edition

General Electric Co. hasn’t required its employees to get vaccinated, noting that such a move would be complicated for its global operations. The company started to educate and encourage workers early in the year to get vaccinated, including working with healthcare providers to ensure easy access to the vaccine.

“At this point, we have no plans to make vaccinations mandatory,” Chief Executive Larry Culp said Tuesday.

In a note to employees Tuesday, Washington Post CEO Fred Ryan said the media company would mandate the shots. “Our plan is to require all Washington Post employees to demonstrate proof of full Covid-19 vaccination as a condition of employment beginning with our September 13 office return.”

On social media, employees’ opinions are split between those who think companies should mandate vaccines at their workplace and those who believe employers shouldn’t be able to force people to get the shot.

Much of the vaccine-mandate debate centers around approval from the U.S. Food and Drug Administration, employment lawyers say. As it stands, shots made by Pfizer, Moderna and J&J have emergency authorization from the FDA, but not full approval. The FDA’s regulatory review process is ongoing and could run into 2022.

“Some employers may feel like they don’t want to force employees to get a vaccine if it’s still in emergency-use phase,” says Jennifer Merrigan Fay, an employment-law partner at Goodwin Procter LLP. Employees and their attorneys could claim that a company is forcing them to take an experimental drug, she adds.

Office workers entering the Goldman Sachs building in New York last week.

Office workers entering the Goldman Sachs building in New York last week.

Photo: Michael Nagle/Bloomberg News

“Once the vaccines have been finally approved, it would reduce the number of arguments that employee lawyers are making that challenge mandatory vaccination policies,” she says.

A Willis Towers Watson survey in May of 660 U.S. employers showed 72% said they wouldn’t require employees to get a vaccine. Since then, some employers have said they are rethinking mandates as the highly contagious Delta variant spreads, says Jeff Levin-Scherz, the firm’s leader on population health.

The Equal Employment Opportunity Commission issued updated guidance in May stating that federal law doesn’t prevent an employer from requiring workers to be vaccinated. But the guidance isn’t a complete green light, says Jim Paretti, an attorney with Littler Mendelson PC’s Workplace Policy Institute.

The American Disabilities Act and Title VII of the Civil Rights Act, which bar discrimination based on religion, race and sex, requires employers to provide reasonable accommodations to employees who aren’t vaccinated because of a disability or religious belief. Actions against a pregnant employee who refuses to get vaccinated could be considered discrimination, Mr. Paretti says.

Companies with national operations could run into difficulties because several states have pending vaccine-related legislation.

“Montana, for example, has a law that says you can’t mandate vaccinations in the workplace when use is allowed under emergency-use authorization,” says Ms. Fay, adding that similar legislation is pending in Illinois, Idaho and North Carolina. “It’s something that employers, particularly large employers and employers with employees in many states, worry about.”

In Texas the governor has prohibited any state agency or organization that receives state funding from mandating the vaccine for its workers. This includes MD Anderson Cancer Center, part of the University of Texas system, which employs more than 22,000 people and treats thousands of immunocompromised people each day. The hospital said 81% of its staff have been vaccinated so far, and MD Anderson continues to run vaccination drives for its employees.

By contrast, a federal judge in Texas ruled last month in favor of a major private hospital system in Houston that mandated the vaccine, triggering a lawsuit that dozens of employees joined. After the ruling, more than 150 workers of the Houston Methodist Hospital system were fired or resigned. In the wake of that ruling, more health-related employers are pressing forward with mandates of their own. In North Carolina, Novant Health, which has 35,000 employees who provide care at nearly 800 locations across several states, announced internally last week that it will require all team members to be fully vaccinated by Sept. 15.

—Thomas Gryta, Sharon Terlep, Benjamin Mullin, Ken Thomas and Aiyana Ishmael contributed to this article.

Write to Ray A. Smith at ray.smith@wsj.com