“This trial is our best shot at getting the vaccine as quickly as possible,” said Ms. DiFransico, who said she wanted Sybil to be able to spend more time with other children and extended family. “We want some semblance of normalcy for her.”

Many parents across the US are racing to put their children in Covid-19 vaccine trials to try to protect them from the virus sooner rather than later.

The shots haven’t been proved to work safely for youngsters, and child volunteers in later-stage trials may get a placebo, rather than the vaccine. Yet parents said they were willing to take the chance and submit their children to the trials’ blood draws, Covid-19 tests and appointments to safeguard them, as well as their family and classmates.

The contagious Delta variant and start of a new school year have added to the urgency, parents said.

Children under age 12 are the last group in the U.S. that doesn’t have access to an authorized Covid-19 vaccine. The youngsters, though at relatively low risk of severe disease and hospitalizations, are going to the hospital in greater numbers than earlier in the pandemic because of Delta’s spread primarily among unvaccinated people.

Partly because of Delta’s spread, many parents have expressed frustration that vaccines aren’t authorized for children under 12.

Pfizer Inc. has said it expects early results from a study of the vaccine it developed with partner BioNTech SE in 5- to 12-year-olds by the end of this month and will seek emergency authorization in early October.

Data for children as young as six months could come as early as October, Pfizer has said, though researchers at some trial sites said it could take longer because enrollment has been slower in that age group than with older children.

Moderna expects to seek emergency-use authorization of its vaccine for 6- to 12-year-olds by the end of this year, and by early next year for children 6 months to 6 years, a company spokeswoman said.

Some parents have been so eager to protect younger children that they pressed to get vaccinations for them after the Food and Drug Administration issued full approval for Pfizer and BioNTech’s Covid-19 vaccine, even though the label specifies use only for people ages 16 years and older. The FDA said last week it was “working around the clock” to help make Covid-19 shots available for children under 12 but said that the testing and regulatory process needed to play out.

Some parents have also sought out research sites seeking young volunteers to test the shots, though not all have been successful. More than 3,000 families contacted Vanderbilt University Medical Center to volunteer for about 150 slots in Moderna’s pediatric trial, said C. Buddy Creech, an infectious disease specialist at VUMC. The strong interest means the medical center turns away most parents, but contacts them from a wait list when spots open up, said Dr. Creech, who is helping lead research for the trial at the site.

Some parents who were successful said they felt reassured that their children had a chance of getting the vaccine. In June, Robby Lindemann of Erie, Pa., enrolled his son Niko, then 7 years old, and his 11-year-old daughter Neveah in the Pfizer trial at nearby Saint Vincent Hospital, part of the Allegheny Health Network. Now, he said, he feels more at ease with them attending school in person as the contagious Delta variant circulates.

To become more assured the shot was safe, Ms. DiFransico said she waited for Pfizer to finish early testing of its vaccine in a small group of children before enrolling Sybil in the pivotal study of more than 4,500 children.

A few weeks later, earlier this month, she signed up her 5-year-old daughter Charlotte in an effort to protect her should she transition from virtual kindergarten to in-person this winter.

“This helps our kid get safe as soon as possible, but it also helps other kids get as safe as possible,” said Ms. DiFransico, a former adjunct college professor now staying home with her children, who said she was aware her children might receive a placebo.

A combination of personal and public reasons drove the interest, according to researchers studying the vaccines and parents placing their children in the trials. The parents wanted to help the scientific community answer whether the shots worked safely, while getting a head start on protecting their children.

“I know the science is sound enough that I’m willing to let my kids be in the trial to make sure that it’s safe for other kids,” said Dr. Sameer Bakhda, an emergency-room physician from Monterey, Calif., whose sons Ajay, 9, and Kian, 8, enrolled in the Pfizer trial at Stanford Medicine in August.

Some families were especially eager to get their children in the studies because the parents hold jobs in healthcare, education and other lines of work that raise their risk for exposure.

When two spots opened at Duke University in August, Sonali Bracken accepted them immediately over the phone for her 3-year-old son Zavian and 9-month-old daughter Kareena. She scheduled appointments for the following week.

“As soon as they could get something in their arms, I wanted them to do it,” said Dr. Bracken, an immunologist and rheumatologist at Duke. “I wanted my kids to have a chance to get the vaccine as soon as possible, especially knowing that the likelihood of it rolling out for the masses wouldn’t be until maybe late fall, early winter.”

Pierce Sandwith, a lawyer from Nashville, said he enrolled his 2-year-old daughter Caroline in the early-stage Moderna study in June, less than an hour after VUMC researchers notified him a spot was available because of a cancellation.

Every child in the study was vaccinated, because the trial was exploring the dosage to use. Mr. Sandwith expressed hope that Caroline, now that she is vaccinated, will better protect her 4-year-old sister, Louise, who is receiving treatment for leukemia, is at high risk for Covid-19 infections and wasn’t eligible to enroll in the study.

“It seems like we’re not going to get out of the pandemic unless children are vaccinated,” he said.