However, North Shropshire MP Owen Paterson, who has spoken to Covid Vaccine Minister Nadhim Zahawi, says he has been reassured the four top priority groups will still get their first vaccine dose by the middle of February.
It comes after NHS England figures showed a total of 27,764 doses of the coronavirus vaccine were given out in the county between December 8 and January 17 – the lowest in any area of England.
However, health bosses said it included almost half of residents aged over 80, and there are a number of other vaccination sites coming on line in the next few weeks.
They are expected to include Telford International Centre, Ludlow Racecourse, the indoor bowling centre at Shrewsbury Sports Village and Whitchurch Civic Centre.
Shropshire Council says that from the end of next week, eligible patients registered at all Shropshire GP practices will be able to receive Covid-19 vaccinations from GP-led vaccination centres.
Mr Paterson said he had a constructive call with Mr Zahawi today and is expecting the situation to improve as the AstraZeneca vaccine is rolled out and further GP practices come on board to administer it.
He said: "I think it's mainly due to the problem of delivering the Pfizer vaccine due to the low temperature it has to be kept at.
"He's aware of the problem. It does make it difficult.
"That will change soon. He's on top of it.
"He's clear as we bring in the AstraZeneca vaccine, which is easier to move around, by bringing in GP practices in Whitchurch and Market Drayton we will have a much closer delivery to people.
"He is clear we will meet the target to deliver for the four priority cohorts by mid-February."
Mr Paterson said that by the end of Sunday, all over 65s in care homes in Shropshire are expected to have received the first dose of the vaccine.
It comes as further talks took place this afternoon between MPs and health leaders in the county.
Dean Carroll, Shropshire Council’s cabinet member for adult social care, public health and climate change, said it was "appalling" that fewer doses had been given in the county than anywhere else in England, but he also said it was good news the vaccination programme is accelerating.
The council says it is working closely with local NHS colleagues and other key organisations to support the delivery of the vaccine.
Its customer contact centre has made over 5,000 calls to frontline health workers and residents to help book their vaccination appointments.
It has also offered support to help people with transport to the vaccination centres where needed, with delivering the IT infrastructure to the centres, project support, liaising with care home teams and volunteer recruitment.
People are being asked to wait until they are contacted by the NHS and are invited for an appointment.
For more information about vaccination roll-out, go to www.gov.uk/government/collections/covid-19-vaccination-programme