But some people are becoming agitated waiting to hear when they will have their jab, health bosses have been told.
A meeting of Shropshire, Telford & Wrekin Clinical Commissioning Groups' governance boards heard yesterday that some inquiries were "quite confrontational in nature" and people are being asked to be patient until they are contacted by the NHS.
Director of corporate affairs Sam Tilley said the county had taken delivery of the Oxford AstraZeneca vaccine, but added: "It is not in the volumes that we would have necessarily hoped for in these early stages.
"We are expecting that to ramp up. That along with the additional sites coming on line will mean we can expand the programme significantly."
There are a number of sites in the county where the vaccine is being given, with more to be announced soon.
Ms Tilley said Woodside Pharmacy in Telford will also be commencing vaccinations imminently.
She said: "It will take a considerable amount of time to work through everybody who is eligible for a vaccine.
"I would ask that people are patient. We are aware of the frustrations.
"We are getting some inquiries that are quite confrontational in nature. That is quite difficult for staff to manage.
"People are working flat out around the clock to make sure this programme is rolled out across the county as quickly as possible."
Dr Julian Povey, chair of Shropshire, Telford & Wrekin CCG, said vaccines were being given soon after they arrive in the county.
He added: "We've done well over 10,000 vaccinations and the number is increasing on a daily basis as more services come on stream."
The governing body was told there had been 'agitation' from people around communication regarding the vaccination process.
David Evans, accountable officer for the CCGs, said: "There is a slight challenge around this in that the comms messages are being quite tightly controlled from the centre.
"We've been asked not to share in public what the level of vaccination rates are.
"I personally don't think that's particularly helpful and I would like to be able to say much more in public, because I do think we need to give that assurance to members of the public.
"There clearly are some key messages, one of which of course is obey the rules.
"We have got a vaccine, that is light at the end of the tunnel we hope, but it is a very long tunnel at the moment."
Liz Noakes, director of public health for Telford & Wrekin, said to give assurance and confidence to the public the NHS needed "to come clean in terms of how the vaccine is being rolled out and what the numbers are".
She added: "I think the NHS needs to be a lot more open and transparent within this area going forward."