Auckland Co-op Taxis says it will now give priority to Stevie-Ray.
CCS Disability Action's disability action team leader Susan Sherrard said Stevie-Ray's experience reflected a region-wide shortage of disability taxis.
"It's just an awful situation," she said. "I've known stories of people who have missed planes because accessible taxis haven't arrived."
Another wheelchair user Dr Huhana Hickey said there was "a severe shortage" of accessible taxis across Auckland.
But Auckland Transport spokesman Wally Thomas said his agency stopped paying for taxi hoists "three or four years ago" after a survey found that there were 161 vehicles with hoists in the region.
"It was considered that there were more than enough hoists in the region to service the clients who needed it," he said.
Auckland's population grew by 121,200 in the three years to last June.
Stevie-Ray, a prop and second-row forward for Kawerau's Putuaki Stags league club, did not know he had terminal cancer until he ruptured the tumour on a bouncy castle at the club's prizegiving late last year.
"That's when they found out there was something wrong with him," said his mother Corinna Tiopira.
Whakatane Hospital sent him to Tauranga Hospital for a scan, and while the family was driving back to Whakatane the hospital rang and told them to turn around and take him straight to Starship in Auckland.
Corinna, Stevie-Ray's dad George Tiopira and his 2-year-old sister Poniwahine have stayed with him in Auckland ever since, staying first at Ronald McDonald House at Starship, then at the Wilson Centre, and in a motel paid by Work and Income since the beginning of last week.
A Wilson Centre social worker is trying to find social housing for them but has been unable to find a wheelchair-accessible home in Auckland.
Stevie-Ray's entire league team visited him in Starship before Christmas. His school, Kawerau South, has raised money for him. Ronald McDonald House chose him to draw the name of Venus Williams's opponent in the ASB Classic tennis tournament in January.
But the family has had to start a Givealittle page to try to buy a van and hoist themselves because the taxi service is so unreliable. They could buy a secondhand van for $25,000, and so far they have raised $670.
Corinna Tiopira said Stevie-Ray loved school and hated having to sit in the motel room all day.
"We sit around because we can't take him in our car any more, so we take turns staying with him," she said.
Auckland Co-op Taxis chairman Jacob Patel said the co-op and its affiliates had 13 or 14 disability taxis across Auckland, including three owned by the group directly. But the others were owned by individual drivers, and cost $40,000 to $50,000 for a van plus $18,000 to $25,000 for a hoist.
He said the Auckland Transport decision to stop funding the hoists made it hard for individual drivers to buy new disability vans. But he said the co-op would improve its service to Stevie-Ray.
"Today I spoke to the operations manager and said `what can we do', it's not right, we like children to get to school on time," he said.
"This particular job now you can be sure that's been taken care of."
Waitematā District Health Board's general manager for child, women and family services Stephanie Doe said the board was working closely with the co-op and the Health Ministry to meet the family's needs.
"Waitematā DHB's service at Wilson Centre was very concerned to learn that transport issues had prevented a patient from attending appointments and schooling," she said.
"We appreciate this is an already difficult time for the family and we are doing everything possible to support them and ensure that there is reliable transport in place."
