The Oxford-AstraZeneca Covid vaccine is being rolled out to hundreds of GP-run vaccination sites in England.
As part of the biggest vaccination programme in NHS history, the aim is to offer jabs to most care home residents by the end of January.
By mid-February, the target is to vaccinate 13 million people in the top four priority groups.
But one surgery visited by the health secretary to promote the initiative said its delivery had been delayed.
And leaders in Birmingham have warned the city could run out of the Pfizer/BioNTech vaccine on Friday and that it has not received any stocks of the Oxford/Astra Zeneca jab.
A letter sent to Health Secretary Matt Hancock, signed by the Labour leader of Birmingham City Council, Ian Ward, Labour MP Lam Byrne and Conservative MP and former minister Andrew Mitchell, said there was “currently no clarity on when further supplies will arrive”.
They also criticise a lack of clarity over the vaccination programme, and say “it remains unclear who is responsible for overseeing the vaccination programme in Birmingham, and whom we should hold accountable for progress and delivery”.
More than 700 local vaccination sites will administer the Oxford jab.
Another 180 GP-led sites, 100 new hospital sites and a pilot scheme involving local pharmacies will open this week.
The vaccine was initially given to patients in selected hospitals, including first recipient 82-year-old Brian Pinker.
Mr Hancock said the vaccine was now being supplied to GP practices across the country as he visited the Bloomsbury Surgery in central London.
The health secretary said he was “delighted” care homes residents would begin receiving their first Oxford jabs this week.
“This will ensure the most vulnerable are protected and will save tens of thousands of lives,” he said.
But GP Ammara Hughes, a partner at Bloomsbury Surgery, told broadcasters its first delivery of the jab had been pushed back 24 hours to Thursday.
She said: “It’s just more frustrating than a concern because we’ve got the capacity to vaccinate. And if we had a regular supply – we do have the capacity to vaccinate three to four thousand patients a week.”
Mr Hancock said the “rate-limiting” factor in efforts to get people vaccinated was supply from the manufacturers.
The surgery said it was continuing to administer the Pfizer/BioNTech vaccine and had so far received three deliveries of that jab.
Labour leader Sir Keir Starmer said vaccination centres and GP surgeries needed better information about deliveries of the vaccine and how much to expect to arrive.
Shadow health secretary Jonathan Ashworth tweeted that the scenes at the surgery were like something from political comedy The Thick of It, but added: “Sadly it’s no laughing matter.”
The Pfizer-BioNTech vaccine was the first to be approved by the UK’s medicines regulator in early December, followed by the British-made Oxford-AstraZeneca vaccine four weeks later.
Because it does not need to be stored at ultra-cold temperatures like the Pfizer-BioNTech vaccine, it can be transported and stored more easily, making it simpler to vaccinate housebound people and those in care homes.
Around half a million doses of both vaccines are ready to be used this week, with millions more in the pipeline in the coming weeks.
Next week, seven major vaccination hubs across England are set to begin operating, including the Excel Centre in London and Millennium Point in Birmingham.