One of the biggest unknowns is who will be responsible for covering the cost of testing for employees who choose to remain unvaccinated, lawyers said. Under the mandate announced last week, all employers with 100 or more workers would have to require that their workers be vaccinated or undergo at least weekly Covid-19 testing. Employers that don’t comply can face fines of up to about $14,000, according to the administration.

Companies also are wondering exactly how OSHA will determine the threshold for which businesses are covered by the requirement, and about exemptions for employees who continue to work from home or who have religious or medical accommodations that are protected under the Americans with Disabilities Act.

The Biden administration will also require federal employees in the executive branch and government contractors to be vaccinated against Covid-19. The new mandates will cover about 100 million workers, or two-thirds of all workers in the U.S., administration officials said.

Although some companies already had vaccine requirements or were considering them before President Biden’s announcement, the move puts pressure on many others to begin determining how to comply with a rule that hasn’t been fully articulated and that is likely to face a flurry of legal challenges and conflicts with state laws.

The first question many companies will face is whether they want to allow employees to opt out of getting a vaccine and instead undergo weekly testing, employment and compensation lawyers said. The logistics and costs of providing such tests to what could be a sizable percentage of a company’s workforce could pose serious challenges for some businesses. And the federal government already has made it clear that companies can require vaccines for employees physically entering a workplace, with certain exceptions.

“Do we want to even bother with the weekly testing option, knowing how much trouble that’s going to be and how many issues that’s going to open with expense, and whether insurance is going to cover it?” asked Robin Shea, an employment lawyer at law firm Constangy Brooks, Smith & Prophete LLP.

Companies will have to decide whether to offer their own testing services through a private healthcare provider or direct employees elsewhere. Setting up testing could be an administrative challenge for large companies in rural areas, in particular, Ms. Shea said.

Businesses with hourly employees who are covered by the Fair Labor Standards Act likely will have to pay those employees for the time they spend getting tested, lawyers said.

As for the costs involved, larger companies might have more financial flexibility to pay for testing programs, while smaller ones are often beholden to what their health insurers are willing to cover. Insurers don’t have to cover workplace Covid-19 testing that an employer does to broadly screen for infection, according to federal guidance, only tests a doctor orders to diagnose or treat a patient, or tests given to someone who has been in contact with an infected person.

That could leave it up to OSHA to clarify whether employers or employees are on the hook for testing costs. Some lawyers said leaving workers to pay for their own tests could work as an effective incentive, alongside benefits surcharges and other measures that effectively penalize employees who don’t want to receive a vaccine. Others said they find it unlikely the government would allow companies to place the costs on workers.

The problems posed by offering a testing option could be partly by design, said Robin Samuel, an employment lawyer at law firm Baker & McKenzie LLP.

“Biden’s goal is to encourage vaccination,” Mr. Samuel said. “So testing is an out from vaccination, and I think they are going to make it more difficult for employers to choose the testing option as a result.”

The coming OSHA rule has shifted the debate from whether companies should require vaccines to how to implement the government’s mandate. For some companies that were reluctant to impose such a requirement on their workers, the president’s administrative action might be a relief, since it provides them with cover, Mr. Samuel said.

The biggest fear for companies considering mandates has been the potential for employee backlash and workers quitting. Ms. Shea noted backlash against vaccine mandates in the healthcare sector, where such requirements are common, in the form of resistance and protests.

“Employers are definitely going to have to sort out a lot,” Ms. Shea said. “Is this going to be a nightmare or a blessing for employers? Because they can now say, ‘The federal government is making us do it—it’s the law and we have no choice.’”

Write to Dylan Tokar at dylan.tokar@wsj.com