Six-year-old Holly Vincent is blind, quadriplegic and has cerebral palsy and epilepsy.
Her mother Riven told the Daily Mirror they cannot get the money for extra care at home despite reassurances from Mr Cameron during a visit to the family's Bristol home last April.
"He promised us personally before the election that my Holly and the NHS would be safe in his hands.
"But clearly he has broken his vow and we feel very upset," she said.
A Downing Street spokesman said: "The Prime Minister is very concerned at what he's heard. He's writing to Riven Vincent and in conjunction he's also asked her local MP for all the details of the case urgently.
"He's also asking the MP to speak to the local council to make sure that she's receiving all that she's entitled to."
Mr Cameron's son Ivan was severely disabled and suffered from cerebral palsy and severe epilepsy. He required intensive round-the-clock care and died in 2009 aged six.
Ms Vincent and the girl's father Mark, a £22,000 a year part-time research scientist, said they receive just six hours a week respite with a carer but will not be given any further help.
The 41-year-old mother, who suffers from MS, said Mr Cameron had "let us down badly and it's affecting vulnerable families like us right across the country".