One woman whose fiance died after the jab was awarded £120,000 this week.
BBC News has since learned two more people have been told they will receive payments.
But many more are still waiting for their cases to be assessed, despite some having final death certificates meaning senior doctors and lawyers have concluded the vaccine caused their loved one's death.
As of May, more than 1,300 claims had been made to the Vaccine Damage Payment Scheme (VDPS) but only 20 referred for medical assessment.
Meanwhile, some fear their genuine but rare cases are being drowned out by a flurry of people making unproven claims about vaccine damage online.
Vikki Spit, from Cumbria, became the first person to be awarded money, following the death of her fiance Zion, who developed vaccine-induced immune thrombotic thrombocytopenia (VITT) eight days after having the jab.
VITT causes a very unusual combination of blood clotting and low platelets (part of the blood that causes clotting), leading to excessive bleeding. An even smaller number of these cases can be fatal.
There have been 444 cases of blood clots with low platelets out of 49 million AstraZeneca (AZ) doses given, and 81 deaths, although we don't know that all were caused by the vaccine.
The risk of a healthy but unvaccinated 40-year-old man dying if he catches Covid is about 500 times higher.
But while there is no doubt the vaccines have been lifesaving on a population level, affected families are frustrated as they feel the small minority for whom the AZ vaccine has caused harm are being ignored.
An interim death certificate last summer suggested the vaccine caused Zion's death, Ms Spit says, but it took another year for the government to agree a payment.
And meanwhile, some of those waiting have lost their ability to work or their household's main earner or had to pay for funerals or home adaptations.
Charlotte Wright, whose NHS clinical psychologist husband Dr Stephen Wright was one of the first to have this kind of reaction to the vaccine, says she has been using food banks since he died.
Even the maximum payment of £120,000, which is not guaranteed, would be "nothing compared to what we've lost" in earnings, she says.
Although even uncontested medical-compensation claims commonly take up to two years - and much longer in complex claims - other countries, such as Norway, paid out the first people injured after vaccination in July 2021.
Rare but severe reactions "ought to have been anticipated" and planned for so the scheme could respond "as quickly as it was needed", Sarah Moore, a lawyer advising affected families, says.
Payment levels have not been reviewed since 2010, she says, with the scheme essentially unchanged for decades - despite a cross-party group of MPs warning in 2015 the legislation was "now out of date and should be reformed".
Far from increasing hesitancy, Ms Moore says, a faster and more generous scheme could increase confidence in the vaccine, by reassuring people they will be looked after in the unlikely event they have a bad reaction.
"I have spent 20 years trying to operate this legislation - and I can tell you it doesn't work," Peter Todd, a lawyer specialising in vaccine injury cases, says.
Claimants have to prove they are at least 60% disabled, he says, a rule borrowed from old compensation schemes for industrial accidents such as crush injuries in mines.
But this is very difficult to apply to an autoimmune reaction that might not have resulted in the loss of a limb but may have caused severe fatigue, pain and brain fog.
"It's really just totally inappropriate for what it's now used for," Mr Todd says.
A Department of Health and Social Care (DHSC) official said it was working to "improve the VDPS, to make the process simpler, swifter and more accessible for claimants... with clearer and more regular communication on the progress of their claim".
But the scheme was "not designed to cover all expenses associated with severe disablement" and claimants may also have to access existing government support systems such as universal credit.
Claire Hibbs was unable to work for a year after developing VITT, struggles with chronic fatigue, migraines and brain fog and fears her job could be at risk - but believes she will not be considered 60% disabled.
Like others in the group, she has been upset by suggestions she might be opposed to vaccines - "it's a pro-vaccination campaign," Ms Moore says.
But Ms Hibbs acknowledges false claims about damage from Covid vaccines have been widely circulated online - and research suggests such claims can increase vaccine hesitancy and put people's lives at risk.
Members of the group, Vaccine, Injured, Bereaved UK (VIB UK) have all received official confirmation of a link to the vaccine.
But underneath many of its factual posts, other accounts share reams of false and misleading claims about the vaccine
Ms Spit says she has also been sent a deluge of anti-vax links.
"I don't want to be associated with that," she tells BBC News.
Ms Hibbs is also concerned about online groups where people claim to have experienced vaccine injury despite not having been diagnosed, even asking for money.
And those spreading false or unproven claims are making it harder for genuine cases to receive recognition, she believes.
"Every single ailment that everybody's got over last year, all of a sudden, it's to do with vaccine," Ms Hibbs says.
"And I think a lot of people have got to understand millions of people every year are diagnosed with illnesses."
"It doesn't help us."