The appeal came as thousands of protesters gathered outside Scotland Yard to demand justice for the 24-year-old, who was due to become a father within months.
The Independent Office for Police Conduct (IOPC) has launched a homicide investigation into the incident, but his relatives said they were “shocked” it took until Friday to announce the probe.
In a statement issued on their behalf, Daniel Machover, head of civil litigation at solicitors Hickman & Rose, said: “The family now await the outcome of that investigation, but seek a charging decision in this case in weeks or a few months, not years.”
He added: “In the meantime, the family demand that the Commissioner of Police of the Metropolis immediately suspend the firearms officer, pending the outcome of the investigation.”
Crowds marching to the Met’s headquarters were led by members of Mr Kaba’s family including his cousin, Jefferson Bosela, who said: “We deserve the opportunity to look his killer in the eyes and ask why? Why did you take him… after he spent so little time on earth… when he was sat there in the car, defenceless, unarmed, scared, terrified?”
Stormzy, Hackney Labour MP Diane Abbott and Streatham Labour MP Bell Ribeiro-Addy were among others to turn up to the protest to show support for Mr Kaba’s family.
The musician told demonstrators: “Chris has a mother, he has a family, he has brothers, he has friends, people who knew him in real life, who for them it’s unbearable.”
He said there was no way to “sugarcoat” what had happened to Mr Kaba and urged them to “have stamina” in their efforts to raise awareness of his death and the events surrounding it.
“Everyone here today, I would encourage everyone to have stamina, and I know it’s a very difficult thing to say… But when these people do these things, they get away with it, because what happens is we do this once, we get tired, we tweet, we get tired, we do it for a week, we do it for two weeks, we do it for a month, and they know we get tired,” he said.
“What they’ve done is they’ve killed someone. We can’t sugarcoat it,” he added.
Protesters paid tribute to the 24-year-old, who died after a police chase that ended in Streatham Hill, and held up signs reading “Black Lives Matter”, “Justice for Chris Kaba” and “Abolish the Met”.
Speakers using a megaphone led chants of “no justice, no peace” and “police are the murderers”.
There was a limited police presence as demonstrators assembled on Parliament Square and made their way through Whitehall.
The IOPC has previously confirmed that no gun was found in the Audi that Mr Kaba was driving.
The PA news agency understands the only firearm recovered from the scene was the one involved in the shooting.
Mr Kaba died after his car was hemmed in by two police vehicles in narrow residential street Kirkstall Gardens and one round was fired from a police weapon.
He was given first aid by officers at the scene and taken to hospital where he died later in the night.
His family, in a statement issued on Wednesday, questioned whether his life would have been “cut short” if he were not black and demanded “answers and accountability”.
Jessica Clayton, 30, a nursery worker attending the protest, was emotional as she described feeling “physically ill” after hearing of Mr Kaba’s death.
Speaking outside Scotland Yard, she said: “If we’re not here for each other nobody else is going to be here if we don’t come out and actually voice what’s going on.
“It’s heavy. I’ve been very ill – physically ill – since I heard. It’s a heavy feeling. It’s unfair.
“You expect questions to be asked before somebody is killed.”
Nathaniel Martin, 30, said he was “frustrated and upset” that “at any time men like me and Chris can have this happen to them”.
Mr Martin, a researcher from London, said he was attending the protest for Mr Kaba outside Scotland Yard to show “solidarity with the family” and “hold the police to account”.
“I think we need to show there are people out here that care about this,” he said.
“I’m really upset, I’m really angry – I’m frustrated that at any time a man like me or Chris can have this happen to them.
“He didn’t need to die.”
Morgan Williams, 21, said young black people should be able to live their lives without feeling threatened.
Ms Williams, a student from London, said she feels unsafe in the wake of Mr Kaba’s death.
Speaking outside Scotland Yard, she said: “I feel like in 2022 we shouldn’t be having these problems.
“As young black people we should be able to coexist alongside other communities without being threatened.
“Of course I feel unsafe. I feel like there hasn’t been an answer, nor has anyone taken responsibility.”
Asked about the homicide investigation into the shooting of Mr Kaba, she said: “I think (a charging decision should be made) quicker than in months and I think people should be brought to justice very quickly – I feel like if it was a white person there would be a lot more being answered for but because it’s a black person it doesn’t seem to be as important.
“Being someone who is half white and half black is hard for me – I feel like half of my identity is still traumatised on a daily basis by the other half of my heritage.”
Met Assistant Commissioner Amanda Pearson said on Friday the officer involved is not currently on operational duties and their work status will be carefully considered going forward.
She said the force was cooperating fully with the IOPC investigation, adding: “I absolutely understand that this shooting is a matter of grave concern, particularly for our black communities.
“I also know what a difficult and often dangerous job firearms officers in particular do every day to try to protect the public.
“They understand and expect that on the very rare occasions they discharge their weapons they will face intense scrutiny.
“I don’t underestimate the impact on them of this development.”