Banbury Primary Care Network (PCN) has been holding clinics every day since last week and the area's care homes have been pulling out all the stops to take residents and care staff to surgeries for their injections.
Dr Gwyneth Rogers, a GP and clinical lead at Banbury Cross Health Centre, said: "We are strongly encouraging everyone to get the vaccine done. It's got a really good efficacy; it creates lots of antibodies to the virus - more antibodies than when you get the illness and we hope to be able to get a large chunk of the population immunised and then, in time, back to some kind of normality."
Dr Rogers described the weekend's clinics as a 'big push' and said local nursing homes were 'brilliant', doing all they could to take residents for their vaccinations. It is expected vaccinations will soon be able to be taken into nursing homes for those who could not travel.
She stressed that no one should phone surgeries asking for a vaccination. A central system called Pinnacle is coordinating invitations to those in the first stage of the programme. The system will relay information from clinics back to patients' own doctors.
"We want those most vulnerable done first as they are most at risk. These are over 85 and over 80-year-olds in homes and their carers who sometimes work in more than one home and might potentially transfer the virus. They've been very keen to be immunised," she said.
Dr Rogers said around 1,200 people had been vaccinated over the weekend from a list of 1,643. Staff have worked overtime to ensure all those attending got their jab. More vaccine has been ordered to extend the programme.
But she warned that those having had the vaccine still need to observe Covid guidelines.
"Even with the first vaccine you don't have full protection and must continue to wear a mask, keep your distance, wash hands frequently, avoid touching your face and wash hands after touching frequently touched surfaces like door handles," she said.
"Even young people have died of the virus and everything we can do to protect the public is a good thing. After three weeks, patients will have a second vaccination but even then I think it would be safe to stick to the precautions."
Dr Rogers said: "It was lovely doing the clinics. Some were saying it was the first time they'd been out since March and some were even tearful - in a happy way - about being able to get out. Some are not going to be with family for Christmas but they accept it because of the dangers.
"This vaccine also gives hope for all those people who have been suffering loneliness and stress because of it all.
"The theory is the vaccination programme will carry on til we've got all the population to the age of 16 covered. As with everything to do with Covid we're sorting things out as we go and I think once the the mass vaccine centres are up and running for the more able, it will speed up.
"Please don't call the surgery, we will ring you. We're really excited about this vaccine. It's an amazing thing; I don't know how scientists did it so quickly."
Dr Rogers said patients stay in the clinic for 15 minutes after their injection for observation but there have been no worrying reactions.
"It's been brilliant, great, such a wonderful thing to be part of. There's been a real community spirit among the staff with the vaccination offering some hope after this last year."