Some parents eager to vaccinate their children under 12 against Covid-19 are hoping that the Food and Drug Administration’s full approval of Pfizer Inc. and BioNTech SE’s vaccine could make that happen even before the FDA authorizes shots for emergency use for that age group.

But the FDA and Centers for Disease Control and Prevention have emphasized that the safest thing for this group of children is to wait for more data to be analyzed.

Monday’s decision prompted many parents to call their pediatricians and ask them whether they will administer vaccines for children under 12. The FDA’s approval generally means vaccines are eligible for off-label use, meaning beyond approved populations, but the CDC said Monday they aren’t authorizing that.

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Pfizer’s vaccine is authorized for emergency use for children aged 12 and up and fully approved for those 16 and up. Moderna Inc.’s and Johnson & Johnson’s vaccines are authorized only for those 18 and over and aren’t fully approved.

Parents are reporting mixed success. Some say they have been able to schedule appointments, while others say they have been refused.

Patti Mulligan, 44, says she called her pediatrician’s office Monday as soon as she saw the FDA approval news. She was able to schedule a first dose for her 9-year-old daughter on Wednesday.

“I don’t have any hesitation. I’ve been trying to get my daughter into a trial unsuccessfully. I am just ready to get her more protected as soon as possible,” says Ms. Mulligan, who lives in Raleigh, N.C. She says she has an appointment for her daughter with her pediatrician but will remain on edge until she receives the shot.

The FDA and CDC cited the absence of safety data for the vaccine in children under 12 as the reason for advising against it. The CDC said a different dosage would likely be required and pointed to Pfizer’s plans to amend its emergency use authorization for children aged 5 to 11 in September once it has more results. On Monday, the American Academy of Pediatrics warned doctors and parents against vaccinating children under 12.

The Biden administration announced that Americans who have been fully vaccinated with a two-dose regimen against Covid-19 should receive a booster, citing the threat from the highly contagious Delta variant. WSJ breaks down what you need to know. Photo: Hannah Beier/Reuters The Wall Street Journal Interactive Edition

“We really don’t have any data” on the safety and tolerability of the vaccine on 5-to-11 year olds, says Sean O’Leary, a professor of pediatrics at the University of Colorado School of Medicine and vice chairman of the American Academy of Pediatrics’s committee on infectious diseases. Children under 5 are being studied, he says, with data coming later.

Dosages for 5-to-11-year-olds will likely be much smaller than what is available, he says. Higher dosages administered to those children could cause unexpected safety issues. “We just don’t have data and we need it,” he says.

Pfizer is testing the vaccine in children under 12, down to children as young as 6 months old but expects results for 5-to-11 year olds in September, with other ages after that. Should results show that it is safe and effective, it plans to request full approval. Moderna is also testing its vaccine on children under 12 down to six months, and J&J is testing its vaccine on children 12 to 17.

The CDC issued updated guidance on the Pfizer vaccine Monday, saying doctors can’t offer the vaccine to patients under 12 as part of off-label use, and that any off-label use of the vaccine, such as providing boosters, isn’t authorized.

Providers who offer off-label use of the vaccine may not have immunity from claims because such usage may not be covered under the public-health emergency legislation that shields providers from liability.

They might also not be reimbursed for the costs of administering off-label shots, and administering them might also impact a provider’s ability to remain in the CDC’s Covid-19 vaccination program. All Covid-19 vaccines in the U.S. have been purchased by the federal government, which allows only those enrolled in its program to administer them.

The highly contagious Delta variant of Covid-19 is tearing through the U.S., driving up infections primarily in unvaccinated communities, as children head back to school. More children have become infected since the strain began circulating this summer. Available data suggest that coronavirus-associated hospitalization and death is uncommon in children, according to the AAP.

Health officials remain focused on trying to get more older children who are eligible for the Covid-19 vaccine to show up for shots. According to the AAP, 43% of all 16- and 17-year-olds are fully vaccinated. For 12-to-15 year olds, the figure is 33%.

Monday’s decision changes the calculus for parents such as Kelly Berg.

Ms. Berg, a 45-year-old stay-at-home mom in Los Angeles, has contemplated an early vaccine for her 11-year-old triplets. In Texas, where the family has a second home, “you can bring your kid in to get vaccinated and they don’t really ask questions.” She says that she knows vaccine recipients there under 12 but that she doesn’t want to teach her children that it is OK to lie.

Now, with the news of approval, “if there is a way to get around it legally, I am so there,” she says.

Marisol Gerardo, 9, was held by her mother as she got the second dose of a Covid-19 vaccine during a clinical trial for children in Durham, N.C., earlier this year.

Photo: shawn rocco/duke health/Reuters

Other parents say they tried to request vaccines after Monday’s authorization but haven’t had any luck.

Melanie Stickle, a 35-year-old mother of two in Silver Spring, Md., says that when she heard of the full approval on her parent WhatsApp group, she called her pediatrician’s office to see if they would consider vaccinating her two young children, aged 5 and 7.

“They said they weren’t recommending it and wouldn’t be doing it,” she says.

Regardless of what pediatricians suggest, parents are likely to press the issue in the coming weeks, according to doctors.

Parents have been clamoring for months to vaccinate their children under 12, says Scott Krugman, a pediatrician at Sinai Hospital in Baltimore. He says that while parents might point to other drugs that are routinely used off-label for children, such as acid blockers or asthma medicines, pediatricians “have never really used vaccines in an off-label way.”

A vaccine is different from medicine used to treat an existing illness or condition, he said.

“If you are giving something to someone who is perfectly healthy, the burden is on making sure there is no potential side effect,” he says.

With some evidence of myocarditis, a heart inflammation occurring rarely among vaccinated teens and young adults, he says, it is particularly important to make sure the dosage for smaller children is correct.

“With concerns over the Delta variant, a lot of pent-up demand, and school starting or has started in many places,” he adds, “this is going to be a hot topic.”

Write to Felicia Schwartz at felicia.schwartz@wsj.com and Anne Marie Chaker at anne-marie.chaker@wsj.com