Tom Bleha and his wife were counting the days until their coronavirus vaccine appointments on Jan. 30, and the promise of a return to what he called “a semi-normal life,” but on Friday evening, Virginia Hospital Center canceled their slots.

“We are outraged,” said Bleha, 85, whose two previous bouts with pneumonia put him at high risk for a serious case of covid-19.

The Arlington hospital will run out of vaccine doses in the coming week after Virginia abruptly changed the way it allocates the doses it receives from the federal government. Instead of doses being shipped directly to hospitals, they will now be scattered throughout Virginia localities based on population, upending a distribution system that had been in place for several weeks.

While Arlington County has said appointments, such as those the Blehas had, will be rescheduled, the sudden change is indicative of the problems and uncertainty that have plagued the rollout of coronavirus vaccine doses throughout the greater Washington region since they began arriving at loading docks in mid-December. And those problems could be magnified once more people are allowed to get vaccinated.

By Monday morning, the number of people eligible to receive coronavirus vaccine doses in D.C., Maryland and Virginia will have reached more than 7.5 million — expanding from health-care workers and seniors to teachers, and, in some places, those with high-risk behaviors as common as smoking.

But the region has received only about 1.7 million doses, triggering a race to vaccinate people and manage ballooning waiting lists.

In the District, thousands of people trying to claim limited appointments crashed websites and jammed phone lines, only to find slots quickly filled. Many of those who did get appointments found themselves waiting in long, slow-moving lines.

Dozens of Montgomery County residents, including elderly people and teachers, used a leaked registration link to try to obtain the vaccine before it was their turn, while Gov. Larry Hogan (R) deployed the National Guard to Prince George’s County to help people in that hard-hit and historically underserved community vaccinate residents.

And in Virginia, Gov. Ralph Northam (D) acknowledged that technical problems skewed data reporting, raising questions about whether there were unused or missing vaccine doses.

Hospitals and health departments don’t know how many doses they’ll receive from week to week, making it impossible to open new clinics. Some hospital chief executives and elected officials say they worry there won’t be enough shots to administer second doses in the two-shot regimen, let alone to all the people in the eligible groups.

State officials say they are hopeful the Biden administration will find a way to ramp up production of Moderna and Pfizer vaccine doses and approve new vaccines from AstraZeneca and Johnson & Johnson, but that could take months.

In Virginia as of Friday, hospitals, health departments and pharmacies had administered at least 424,857 doses of the 1 million that the state has received from the federal government. Most of the state on Monday will have moved to Phase 1B, which means on top of health-care workers and long-term care facility residents and staff, people 65 and older, people 16 to 64 with underlying medical conditions and a wide-ranging group of front-line essential workers, including teachers, grocery store workers and postal workers, will be eligible for the vaccine.

About 5.5 million people statewide fit this description, according to the Virginia Department of Health. (Portsmouth plans to move to Phase 1B on Tuesday.)

Alexandria was one of the first localities in Virginia to petition Northam to expand the eligibility pool, but shipments of doses have not kept pace. The city of 160,000 residents has a vaccine waiting list of 20,000 that could grow to 50,000 when new groups become eligible Monday.

“We have this clogged-up drain, and we have the idea to turn on the faucet and dump more water in,” said Alexandria Mayor Justin Wilson (D).

Jeff C. McKay (D-At Large), chairman of the Fairfax County Board of Supervisors, wrote a letter to Northam on Wednesday, requesting more vaccine doses to accommodate a county waiting list topping 100,000.

“We stand ready to expand our distribution to more eligible Fairfax County residents, should the Commonwealth increase our supply,” he wrote. “Unfortunately, one cannot be done effectively without the other.”

Danny Avula, Virginia’s newly appointed vaccine coordinator, said the state has changed the way it allocates doses — directing them away from hospitals and toward local health districts, which are developing a greater capacity to store and administer the doses.

“Now that capacity has ramped up across the state, we really need to prioritize the equitable distribution of vaccine,” Avula said. “Our lack of supply and the increase of capacity of the rest of the state means there just is not enough vaccine to go around.”

The change is jarring for a large health-care system such as Inova, which received 26,000 first doses this past week and will get a fraction of that — 1,175 doses — this coming week.

J. Stephen Jones, a physician and the president and chief executive of Inova, said that his system, which serves more than 2 million people through a network of five hospitals and specialty practices in Northern Virginia, has a new site under construction at the Victory Center in Alexandria, but not enough doses to schedule appointments there.

“It would take at least a threefold [increase] of what our most recent vaccine deliveries have been before we need an additional site,” Jones said.

The Virginia health department has also deployed teams to address technical issues throughout the state, helping providers manually enter data, ensure different computer systems can work together and find better ways to track redistributed vaccine doses, Avula said. He added that the state will regularly post allocations to localities online.

“The criticism is totally understandable,” he said, referring to provider and public complaints about incomplete data. “We don’t have a lot of clarity right now about exactly how many doses have been administered.”

In Maryland, 314,861 of the 702,375 doses allocated to the state had been administered as of Friday, a fraction of the number needed to cover eligible groups.

The state will move to Phase 1C on Monday, expanding eligibility from health-care workers, first responders, people 75 and older and teachers, to adults 65 and older and essential workers in agriculture, manufacturing and the Postal Service.

That adds up to about 2.1 million people, according to the Maryland health department.

“It’s an impossible number,” said Gabor D. Kelen, director of Johns Hopkins Office of Critical Event Preparedness and Response.

Kelen said that even as a large number of newly eligible people can be vaccinated through the Hopkins system, the system is getting fewer and fewer doses as more providers come online.

Meanwhile, Hogan said during a radio interview Friday that the state has 279 National Guard members going into jurisdictions such as Prince George’s to try to help the local health department and other service providers get up to speed.

“We want to get them done a little faster, but we don’t want to run out of supplies like some states have had a disaster on their hands,” he said.

Hogan said health departments “are completely overwhelmed.”

“There’s been no funding from the federal government, no assistance, they just dropped off some vaccines and then said in our state go do 12 million of them,” he said.

In Montgomery County, leaders said they are still working to inoculate 60,000 county residents who are health-care workers and first responders. The county of 1 million, which has received about 7,000 doses weekly, does not yet have adequate supplies to move to the next phase, officials warned.

They are also investigating attempts by residents to jump the line after dozens made vaccination appointments before they were eligible.

Health Officer Travis Gayles put it this way: “We just want to ensure everyone in the county has fair, equal access to the system . . . a system that is not based on who you know.”

D.C. has received 62,000 doses as of Jan. 16, administering about 41,000 of them. The city is vaccinating health-care workers, people 65 and older, first responders and, starting Monday, teachers and police officers — a number that could include 200,000 people, according to city estimates.

District officials have announced new appointment slots, concentrating many of them in the hardest-hit parts of the city, after data showed that residents of less-affected, wealthier wards were getting vaccinated while those in harder-hit areas were making fewer appointments.

Meanwhile, some senior centers had long lines of people waiting to get vaccinated, while others have tried and failed to even get an appointment.

In addition to the upcoming Sibley Memorial Hospital clinics for seniors at 14 public housing sites, the city has also begun opening two batches of vaccine appointments online every week, one for Zip codes with high caseloads and another for residents citywide.

The slots are in high demand: One round was full in 30 minutes.

Mike Johnson, 74, and his wife, Thalia Assuras, who live in the Palisades neighborhood of D.C., tried three ways Friday to register for an appointment — using a phone, iPad and laptop — with no success. They haven’t seen two of their three grandchildren in a year because of the pandemic, and Johnson worries about the virus because he has a history of COPD, high blood pressure, cancer and heart disease.

“You got on at 9 a.m.,” he said, “and at 9:01, everything is gone.”

Correction: This article has been updated to correct the name of Thalia Assuras.

Erin Cox, Antonio Olivo, Gregory S. Schneider, Rebecca Tan, Ovetta Wiggins and Julie Zauzmer contributed to this report.