Though such backlogs are easing, Mr. Bady was seeing firsthand the result of a national shortage of veterinarians and veterinary technicians, one that grew more acute during the pandemic and is expected to worsen in the years ahead.

The demand for pet healthcare services is expected to grow by 33% between 2019 and 2029, according to research cited in a recent report written by

James Lloyd,

an economist and former dean of the University of Florida College of Veterinary Medicine. Given expected retirements and the slow rate at which new vets are entering the workforce, he projected a shortage of 15,000 companion-animal veterinarians, or about 16%, by 2030. The median salary for veterinarians was $100,370 in 2021, according to the U.S. Bureau of Labor Statistics.

“You can see that pretty soon you start to really feel that crunch,” said Dr. Lloyd, who wrote the report for Mars Veterinary Health, a large provider of pet healthcare.

For many pet owners, the crunch became obvious during the pandemic. A wave of people adopted pets for the first time while working from home. Others, spending more time with the dog or cat beside them on the couch, decided to take care of long-term issues that they might not have normally.

At the same time, animal hospitals weren’t able to see as many pets with curbside service. And some vets, experiencing heightened stress from pet owners frustrated by the wait times, either decided to work fewer hours or changed jobs within the industry, said

David Lee,

director of the Center for Veterinary Business and Entrepreneurship at Cornell University’s College of Veterinary Medicine. Some older vets decided to retire with the lure of buyouts from private-equity investors or corporate practices, he said.

“When you put it together, it’s created kind of this perfect storm,” Dr. Lee said.

In addition to making patients wait longer, the pandemic also added to pressure on vets, many of whom graduate with high amounts of student debt and face stress from dealing with the suffering of patients who cannot speak, conflicts over treatment and bills, and ethical dilemmas over euthanasia. Veterinarians often rank high on the list of professions with elevated suicide rates, just behind doctors, dentists and police.

“It’s easier to be less kind when you’re not face to face,” said Sarah Baker, a Chicago vet who is planning to open her own practice in the suburbs next year. “It’s taking care of animals but also there’s a huge piece of customer service. That one bad interaction or one bad review—it just weighs heavily on you.”

Some in the industry think that Dr. Lloyd’s predictions may be overblown, noting that some veterinary colleges have added new slots and a few new schools have recently opened.

“Short term, there’s definitely a supply-and-demand disconnect. But by the end of this decade, we expect that some of that will be smoothed out,” said Dr. Lori Teller, president of the American Veterinary Medical Association. A bigger concern is a persistent shortage of veterinary technicians, she said.

Many individual vets are still working their way through a backlog of patients.

Byron de la Navarre,

chief of staff at Animal House of Chicago, says he is scheduling appointments for well animals six months out these days, compared with one to three months before the pandemic and is trying to hire another full-time vet.

On a recent day, he showed a visitor around his domain, a warren of rooms for exams, processing specimens and procedures. Boarding cats reclined in a kitty condo area; smaller mammals like rabbits and guinea pigs occupied another room, while yet another held cage after cage of exotic birds. Snakes, a tortoise and a bearded dragon lizard hung out in tanks lining the halls.

One of his vet techs had called in sick and the backup also was sick, he said, creating a hectic day.

“We’re busy, which is good, but it’s a lot of juggling,” he said between appointments, callbacks to patients and outside labs, and helping his intern prepare to neuter a cat later in the day.

Sophie Desch, a 25-year-old lawyer, had brought in her cats, Simone de Meowvoir and Caspian, to get an international health certification as she prepared to move them to the U.K. She made the appointment about four weeks ago and was lucky to get the spot since Dr. de la Navarre is one of few vets in the area who has the right training to provide the certificate.

When she brought her cats in six months ago for a routine checkup, the office was still doing curbside check-ins, making communication a challenge, she said. “It was all kind of through a game of telephone, speaking with the front-desk staff who spoke to the vet. I mean things were definitely a little bit more awkward.”

Write to Joe Barrett at Joseph.Barrett@wsj.com