NHS Lothian said the shortage was due to an external manufacturing and packing issue, and had affected a “small number” of GP practices.
The health board was unable to say how many patients had been affected, but said it hoped to receive further supply on Tuesday.
It is understood that among those affected were some housebound over 80s, who were due to receive vaccines from NHS Lothian’s community vaccination team.
The daughter of a housebound 93-year-old in Edinburgh who has yet to receive a vaccination appointment called the needle shortage a “national disgrace”.
"The fact that an extremely vulnerable and elderly lady in her 90s has still received no formal notification of when she might get her first dose of the vaccine, is surely not right?” she said.
"Moreover, the fact that the community vaccination rollout to vulnerable groups has been stopped due to insufficient needles is a national disgrace.”
Her mother was told by her GP practice last week that she would be given a first dose of vaccine by the end of the week, but when this did not happen she called the practice, who told her that the delay was due to a lack of needles.
Her mother’s GP practice had been very helpful, she added, but unable to resolve the issue, which has caused distress to her 93-year-old mother.
"She is so fretful and anxious – it’s heartbreaking to think that she is so late in getting this vaccine and that no one is able to tell me when it might actually happen. All for the sake of insufficient needles,” she said.
Colin Briggs, Director of Strategic Planning at NHS Lothian said, “Due to an external manufacturing and packing issue, a small number of GP practices have experienced recent supply shortages of vaccination needles. More supplies are due to arrive today [Tuesday], which will help to minimise any disruption.
“In any instance where a vaccination appointment has been postponed, alternative arrangements are being sought as quickly as possible.”
As of Tuesday morning, 96 per cent of Scots over 80 not living in care homes had been given a first dose of Covid-19 vaccine.
The Scottish Government target for completion of this group was Friday February 4.
First Minister Nicola Sturgeon said at the coronavirus daily briefing on Tuesday that she could not say that every single person in each priority group would be given a vaccine by that group’s target date.
"There are always going to be some people who maybe haven't got absolute confirmation of their appointment or who might end up waiting longer,” she said.
"That's the nature of a system that is as big as this one in scale, but I think everybody can see how well this programme is going.”
She added that Scots over 75 who have still not received confirmation of their vaccination appointments should email her at [email protected] if they cannot find a solution elsewhere.
"We want everybody to be assured that nobody is being left behind in this vaccination programme,” she said.