Dr Julian Povey, chair of Shropshire and Telford & Wrekin Clinical Commissioning Groups, said he understood people's frustrations with wanting the vaccine sooner, but stressed it was down to supply.
A number of hubs and sites around the county are already offering the jab, with mass vaccination centres also preparing to open – details on their exact locations are expected to be officially announced by health chiefs 'very soon'.
Dr Povey said: "We will have all of our main vaccinations centres open by the end of the month.
"I think there will be two very large ones and then there'll be a number of medium sized ones.
"We would expect to see between 10 and 15 sites in total, again it will be down to supply. We are giving as much vaccine as we are delivered.
"Yes we want it to be done quicker, we want it to be done to more people but the limiting factor is vaccine supply."
He said by the end of this week, vaccinations should have been given at seven nursing homes in the county and by the end of the month it should have been rolled out to all nursing homes.
"I can appreciate people are frustrated and they want it sooner but it's not for want of trying," Dr Povey said.
"We'd all work 24 hours a day if the vaccine was there to give it but it's not there to give.
"In some ways if the Government had waited until they had got enough vaccine to vaccinate everyone and then distributed it, then it would have delayed it, so the Government are in a very difficult position.
"As a system we've had our fair share.
"We can't deliver at this moment to everyone at the same time, no matter how much we want to.
"As soon it arrives it's snapped up and appointments are given and people are vaccinated."
He urged everyone to come forward for the vaccine when it was their turn, with patients being asked to wait until they are contacted by their GP.
Dr Povey said: "We have only done a small percentage of what we will do; clearly when you first start it will look like there are areas that have had more than others, but come the end of the programme everyone who needs a vaccination in Shropshire will have been offered a vaccine and hopefully everyone will accept it.
"It is amazing that we are doing this nine or 10 months after the start of the pandemic in the UK and a year after it was first noticed."
David Evans, accountable officer for the CCGs, added: "This is a huge programme, the logistics behind it are challenging to say the least.
"At the moment we are giving every vaccine we can get our hands on."