ALBANY, N.Y. — Millions of New Yorkers with health conditions that leave them at high risk of illness from COVID-19 can theoretically sign up for appointments at state-run vaccination sites starting Sunday, but a lack of vaccine supply means many will be frustrated in their search for a shot.

Seven million New Yorkers, including health care workers and people over 65, were already eligible for vaccinations under previous state rules. Starting Monday, 3 million more people over 16 with so-called comorbidities will become eligible.

In order to be vaccinated, people will have to provide a doctor's letter, a signed certification or other medical information showing they have an eligible health condition. New York's list includes cancer, pregnancy, heart conditions, weakened immune systems, sickle cell disease, diabetes, neurologic conditions, liver disease, obesity, pulmonary disease, chronic kidney disease and intellectual and development disabilities such as Down Syndrome. 

The website for state-run vaccination sites allowed people to check a box indicating they had an eligible comorbidity starting Sunday, but many attempts to book an appointment yielded a message that said, "Due to high volume, appointments can't be made at this time for this location. Please try again later."

Gary Holmes, a spokesperson for the state Department of Health, said the scheduling site was working "extremely well" and an average of 350 appointments were being booked per minute Sunday.

"As expected, the site is experiencing a tremendous amount of volume after appointments were opened to New Yorkers with comorbidities and underlying conditions this morning," Holmes said in an email. He said the state's vaccine distribution network "is firing on all cylinders, and as the supply of vaccine expands we stand ready to get more shots into people's arms."

New York's vaccination rollout has sped up in recent weeks, but residents across the state have struggled to get appointments, particularly in rural areas that have few pharmacies and are far from state-run mass vaccination sites that can inoculate anyone who's eligible. 

Pharmacies can only vaccinate people over the age of 65, while hospitals have been prioritizing health care workers.

Counties are focusing on eligible essential workers and people living and working in group homes for people with disabilities. But they can also choose to vaccinate taxi drivers, restaurant workers and people with underlying conditions — if they have the supplies. 

Gov. Andrew Cuomo has promised supply will go up now that New York is set to receive 20% more in supply over the next three weeks. That could amount to at least 60,000 more doses each week, based on the state's data. 

New York was set to receive another 317,700 doses intended for use as the first shot in a two-shot protocol for the week ending Saturday — down from 320,525 for the prior week, but up from 250,000 in mid-January.

Many eligible residents haven't received vaccines yet.

New York still hasn't provided a first dose to about 9,500 nursing home residents and 25,000 staff members who've said they wanted to be vaccinated under a federal program that launched in December. The state said it's vaccinated about three-fourths of residents but just half of 131,000 staff at nursing homes that have seen a spike in cases and infections this winter. 

State health officials hope that as more people get vaccinated, New York will start seeing a sharper drop in new COVID-19 infections. 

New York now has the nation's second highest average of daily new COVID-19 cases per-capita over the past seven day, second only to South Carolina. 

Cases are dropping in most states, but are flat in New York as well as New Jersey and Pennsylvania. New York averaged about 8,650 cases each day over the past seven days — essentially unchanged from the prior seven days. 

A winter surge throughout the state that began in November drove COVID-19 hospitalizations up to 8,000 in early January, and as high as nearly 9,300 as of Jan. 19. Since then, hospitalizations have declined overall to about 7,000 as of Friday.

Also on Sunday, Cuomo signed an executive order extending closing times for gyms, casinos and billiards halls from 10 p.m. to 11 p.m. The move came two days after the Democratic governor moved the closing time for bars and restaurants to 11 p.m.

"Hospitalizations and the positivity rate continue to decline, and with the holiday surge now in our rear-view mirror, we must keep taking steps to further re-open the economy safely and responsibly," Cuomo said.