DJ Jo Whiley says it has been "the worst week of our lives" as her sister, who has a learning disability and diabetes, is in hospital with Covid.
The BBC Radio 2 presenter said Frances, 53, had been offered a vaccine on Saturday but it was "too late" and she was now "fighting for her life".
And she urged ministers to "forget the classifications" and protect all those with learning disabilities.
The government said it was working hard to vaccinate those most at risk.
Whiley said her sister had become very ill since testing positive for coronavirus following an outbreak at her care home in Northamptonshire.
"Twenty-four hours ago we were talking palliative care and yesterday she rallied round and we are seeing her oxygen levels rise, so at the moment we have got hope.
"Twenty-four hours ago we didn't have any hope at all, so she is an amazing fighter, she always has been a great fighter and I'm just hoping that her spirit gets her through."
But she said it was particularly difficult to treat people with learning disabilities because they had complex needs and often struggled to communicate.
She said Frances had been so frightened that she would not allow medical staff to give her oxygen, adding: "That is why her oxygen levels plummeted, that is why she ended up fighting for her life."
Whiley said her parents had not slept "for days" but fortunately had been allowed to be with Frances in hospital.
She said: "The idea of Frances having to do this on her own is unthinkable, and also actually people couldn't cope with Frances."
Her sister was so terrified when she was admitted that "she actually rampaged through the hospital and people couldn't contain her and security guards had to be involved, they had to restrain her", Whiley explained.
She added: "The fact that my mum was there and was able to talk her down and to be there with her and to try and give her oxygen was crucial."
Whiley has previously questioned why she was offered the vaccine before her sister, who has the rare genetic syndrome Cri du Chat, but said it was possibly because she was deemed a carer for her sister.
People with diabetes and those with a "severe or profound" learning disability are in priority group six for the coronavirus vaccine, along with unpaid carers for those with disabilities and the elderly.
Speaking to Andrew Marr, Whiley said: "It felt like the cruellest twist in the world because I've been asking for her, wanting for her to have her vaccine for a year... to be protected.
"Then, for me to suddenly get a call to say that I've got the vaccine, it just felt hideously unfair.
"And she was actually called in for her vaccine last night, my mum got a message to say that she could get vaccinated, but it's too late, she's fighting for her life in hospital. It couldn't be crueller."
Whiley said she hoped speaking about her sister would underline the need to get people with learning disabilities vaccinated "as quickly as possible".
She said: "Forget the classifications, forget the cohorts, just protect these people. They are so precious, they're so vulnerable and they have no way of fighting for themselves - we have to fight for them. Get them and their carers vaccinated, stop them dying."
'Huge health inequalities'
Health Secretary Matt Hancock, also speaking to Andrew Marr, said: "My heart goes out to Jo and to her sister and her whole family."
Asked if he found it odd that Whiley was offered the vaccine before her sister, he said it was "very important that we follow clinical advice in terms of the order of prioritisation".
"And in group six, as it's called, that's where we vaccinate both those with learning disabilities and also their carers. And that group is being vaccinated right now."
Edel Harris, CEO of learning disability charity Mencap, said the story of Whiley's sister had highlighted that people with learning disabilities "suffered huge health inequalities" in England and should be prioritised for vaccination.
Ms Harris told the BBC: "The virus is not deciding if you are profound or severe, or mild or moderate - GPs don't even use those definitions. That's why we think they're arbitrary, they don't mean anything."
Nearly six out of every 10 people who died with coronavirus in England last year were disabled, according to figures from the Office for National Statistics.
The Department of Health said in a statement: "We understand this is an enormously challenging period for many people and the NHS is working hard to vaccinate those most at risk, as quickly as possible."
It said the government was following the recommendations of independent experts at the Joint Committee on Vaccination and Immunisation (JCVI) on which groups to prioritise for Covid-19 vaccines.
It added: "The committee advised the immediate priority should be to prevent deaths and protect health and care staff, with old age deemed the biggest factor determining mortality."