The Food and Drug Administration and Centers for Disease Control and Prevention have authorized a third shot of the vaccine made by Pfizer Inc. and BioNTech SE for a wide swath of the US population. Anyone over age 65 is eligible for a booster shot, as is anyone over 18 with a pre-existing condition predisposing the person toward a severe case of Covid-19. Also eligible are those with a job or living situation that poses a higher risk of contracting Covid-19. Boosters for all patients must be given at least six months after an initial Pfizer vaccine course.

Patients who received initial doses of the vaccines made by Moderna Inc. and Johnson & Johnsonaren’t eligible yet. Approval of a booster regimen for those patients is expected in the coming months. The FDA did amend its emergency-use authorization in August, however, to allow a third Moderna dose for immunocompromised people.

Debbie Hirsch, a 67-year-old retired special-education teacher who received the Moderna vaccine initially, wasn’t going to wait; she made an appointment on Monday at her local CVS Health Corp. pharmacy in Poughkeepsie, N.Y. A nurse there gave her a third Moderna shot, no questions asked, she said.

Ms. Hirsch, whose husband is recovering from heart surgery, said she checked a box on the CVS website attesting that she was immunocompromised, even though she doesn’t qualify. FDA guidelines for patients receiving a third Moderna jab include patients who take medication to suppress their immune systems and cancer patients currently undergoing treatment.

“I felt I fell under that category where you’re above 65, and your immunity starts to wane, even though I’m not undergoing chemotherapy or anything like that,” Ms. Hirsch said.

CVS spokesman Michael DeAngelis said the chain is administering third Moderna doses only to eligible immunocompromised people but said it is following CDC guidance to allow patients to self-attest to their eligibility without requiring documents of proof.

West Virginia Gov. Jim Justice said Monday that all adult residents of the state who have been vaccinated for at least six months should seek a booster. “If you’re 18 or above, you will qualify in some way,” Gov. Justice said. “I would really highly encourage you to run to the fire again and get that booster shot.”

A spokesman for the governor said he was encouraging all eligible West Virginians to seek a booster, adding that the CDC’s guidance, “interpreted broadly, allows all healthcare workers, retail workers and anyone who works indoors around other people to qualify.”

White House Covid-19 coordinator Jeff Zients said at a media briefing Tuesday that more than 400,000 people received a Covid-19 vaccine booster at U.S. pharmacies last weekend. Nearly 1 million people have scheduled appointments to get their booster shots, Mr. Zients said.

Valencia Jennings is 27 years old and works in the human-resources department of a hospital in Memphis, Tenn. She says she has no underlying medical conditions but got her first doses of the Pfizer vaccine earlier than most people, in December 2020, because she is technically a healthcare worker.

“I interact with a lot of people on a daily basis, and I don’t know anyone’s vaccination status,” she said. She received her Pfizer booster on Monday from a clinic in the hospital.

Other patients said that pharmacies weren’t asking for confirmation of eligibility beyond their online scheduling tools. Lauch Hines, a retirement adviser from Campobello, S.C., got his Pfizer booster on Monday after calling around to local pharmacies.

“They just said, ‘Come on down and bring your vaccine card,’” said Mr. Hines, who is 75.

Michele Cozadd, a former supply-chain information technologist from Columbus, Ohio, who lives on disability because of a brain aneurysm three years ago, said she got her booster this week without any questions from her pharmacist.

“Even on the scheduling app it didn’t ask me if I had a pre-existing condition or a high risk factor,” said Ms. Cozadd, 46.

People who feel that they are at high risk of serious Covid-19 will do whatever they can to get a booster shot, including skirting eligibility requirements, said Neil Sehgal, a professor of health policy at the University of Maryland.

“Some of the demand we’re seeing for boosters is the shadow of policies we’ve had before that failed to mitigate the spread of Covid,” Mr. Sehgal said. “Thousands of people are still dying each day. We’re not out of the woods yet.”