Riszka's Law would make it compulsory for people seeking prescriptions for the contraceptive pill to have annual check-ups.
Paul Hadley, a 55-year-old musician, and his wife, Melania, aged 51, whose daughter, Riszka Szymkowska, aged 22, died on April 5 last year from a blood clot, said they were "concerned that young women were able to access the pill without annual health checks."
Her family, from Walsall Wood, have now set up an online petition aimed at getting Riszka's Law introduced in her memory.
Paul said his step-daughter, who would have celebrated her 23rd birthday on March 4, had been a support care worker in Bloxwich, and suffered from polycystic ovaries.
He said: "She had difficulties in reaching her doctor because of her shifts and Covid-19 restrictions, and her weight and blood pressure had not been checked for two years.
"The day she died she returned home saying she had a bad back and then complained of a bad leg before going to bed and was later found lying on the floor."
The family have contacted Wendy Morton, MP for Aldridge and Brownhills, about raising a petition but say they need 100,000 signatures for it to be debated in Parliament.
Paul added: "We are not after any kind of compensation but want to save lives and if we only save one it will be worth it."
A spokesman for Walsall Clinical Commissioning Group said: "We are aware of this case and our sympathies are with Mr and Mrs Hadley on the sad loss of their daughter.
"We are supporting the practice as they carry out their investigations and will ensure that any lessons learned will be shared across all practices in Walsall."
By Sue Smith