Her second tweet has been shared more than 8,000 times.
They were inspired by the #MatchForLara campaign that went viral and helped find a donor for another young woman from a mixed-race background.
Zara, from Winchester, Hampshire, opened the Match4Zara account with her father Loua Al Shaikh last week.
He told BBC News: "We thought we had nothing to lose by giving this a try. We put the first tweet out and the notifications on my mobile phone haven`t stopped.
"I had a Twitter account but I`ve never used it. I was amazed at the response we`ve had from all over the world."
Among those who retweeted the appeal were Sherlock actors Mark Gatiss and Louise Brealey as well as American author Rainbow Rowell.
People have started replying saying they will register to see if they could be a donor, including people from mixed heritage.
"It gives us hope and shows Zara that she is not alone."
Zara faces the problem highlighted by the viral #Match4Lara campaign, started by 24-year-old Lara Casalotti who is of Thai and Italian heritage.
It is harder for people with mixed-race backgrounds to find donors because they have rarer tissue types and there are relatively few people of ethnic minority backgrounds registered as donors.
This is partly because of a lack of awareness, says Loua Al Shaikh - who is also a doctor. There is also a myth that the procedure for donation is complicated, when in fact it is quite simple, he says.
As well as the Twitter account, there is a Facebook account called Saving Zara and other family members are setting up a campaign website.
Dr Al Shaikh says he hopes he can get more people registering as possible donors, whatever background they have.
"The more people who register, the higher chance for all waiting for a transplant to find their donor match.
"It`s sad to say, but no-one knows when you or someone you love might need that help."
Zara first went through chemotherapy in 2014 and completed four courses of the treatment - but after 15 months in remission developed acute myeloid leukaemia.
She is currently at Southampton University Hospital after going through another round of chemotherapy which will leave her weak and prone to infection over the next few weeks.
But she is a fighter, her father says. She dyed her hair blue just before her latest round of treatment as a way of taking control of the situation.
"She is also very arty and occupies herself by drawing. She`s big on Manga-style cartoons," Dr Al Shaikh says. "She even learnt some Japanese during her first stay in hospital."
Charity Anthony Nolan says a shortage of registered donors means only 20% of people from ethnic minority backgrounds who need a stem cell transplant will find a perfect match.
Dr Al Shaikh says he is hopeful because Zara`s condition was picked up early - but she cannot keep having chemotherapy. Her only hope for a cure is a bone-marrow transplant.
More about:
