It comes after NHS Lothian advised patients on Friday of a delay to appointment letters being sent out last week, adding that the issue should be resolved by Monday.
Labour MSP for Edinburgh Southern Daniel Johnson told the Scottish Parliament on Tuesday that the letters delay has resulted in some vaccination centres in the Lothians seeing no patients on Monday and Tuesday, with staff told to try and find members of the public or police officers to take the leftover jags.
First Minister Nicola Sturgeon said she was “aware” of the issue, and that it is under investigation by the Scottish Government.
Mr Johnson said: “At the end of last week, it became clear there had been a major problem with the issuing of blue letters, and that people between the ages of 60 and 65 in the Lothians had not been receiving their appointments.
“This resulted in the health board making pleas on social media for people to call and find out if they had an appointment.
“I am informed that this problem has persisted, and that some vaccination centres have had literally no patients over the last few days, and staff have been asked to go out and I'm quoting my source here, ‘drum up support from police stations or anywhere else’ to fill slots.”
SNP MSP Christine Grahame, whose constituency also covers Lothian, said she endorsed Mr Johnson’s concerns about NHS Lothian, and said she had “many of her own.”
Ms Sturgeon said she believes the problem may be due to Royal Mail.
“This is an issue I am aware of," she said.
"We have been investigating it over the past few days, trying to work out, where the issue lies, whether it's with National Services Scotland administering the centralised system, or with Royal Mail.
“There is, we believe, the possibility that it is a Royal Mail issue, because other letters have been arriving, but we are still trying to bottom that out and resolve that.”
Royal Mail said it is working with the Scottish Government to “understand what happened”.
Not all centres in the Lothians were affected, with reports that the Edinburgh International Conference Centre had been “very busy” on Monday.
The total number of vaccine doses delivered by NHS Lothian on Monday was 6,802, more than double those administered the previous Monday.
Across Scotland some 59 per cent of those aged 60 to 64 have received a first dose of the vaccine, while 1,943,507 first doses have been given in total.
Chief Medical Officer Dr Gregor Smith said on Monday that the issue with letters in NHS Lothian did not appear to have affected any other area of the country.
NHS Lothian directed requests for comment to the Scottish Government.
A Royal Mail spokesperson said: “Royal Mail understands the vital importance of getting these vaccination appointment letters to people safely and on time. We are working hard to prioritise this mail.
"Our healthcare partners have already moved towards using blue envelopes to help us identify these letters in the delivery process more easily.
"We are now working with our partners - including the Scottish Government - to understand what happened in this instance. We understand that new appointments will be arranged shortly for any impacted residents.”