The National Blood Transfusion Centre director, Dr Adnan al-Hakimi, said the crisis emerged after French medical charity Medecins Sans Frontieres (MSF) informed the bank it was suspending its aid after more than two years of work.
"We appeal to all humanitarian organizations in the international community and all financial donors to support the center, as our medical supplies have nearly run out," said Hakimi.
An MSF spokeswoman said the charity had handed over its support for the blood bank to the World Health Organization.
"We will only be able to work for one more week, and after that if the humanitarian organizations don't mobilize to support the national center, it will shut down."
The blood bank says it treats some 3,000 Yemenis a month who suffer from cancer, kidney failure, and thalassaemia, an inherited blood disorder that causes severe anaemia.
But a trifecta of war, disease, and famine has left the bank struggling to keep up with spiralling demand, with basic supplies all but impossible to secure in a country locked in by port and airport blockades.
"We've been impacted by the overall situation in Yemen, including the economic collapse," Hakimi said.
Less than half of Yemen's hospitals are still up and running two years into a war between Iran-backed Houthi rebels, who control Sanaa, and a government allied with an Arab military coalition led by Saudi Arabia.
The country's main international airport in Sanaa is also blockaded, with access limited to a select few UN aid flights by the Saudi-led coalition, which controls the airspace.
The war has destroyed much of Yemen's infrastructure and pushed the country - long the Arab world's poorest state - to the brink of official famine.
More than 8,300 people have died in the conflict with another 47,700 wounded and millions displaced, according to the WHO.
A cholera outbreak has also claimed the lives of some 2,000 Yemenis in less than four months.
'What if it shuts down?'
Amina Ali, whose young son has a condition that requires blood transfusions, makes the trip to the center regularly for blood and platelets.
Now, she says, she fears those trips are numbered.
"I order blood and platelets for my son every 10 days from the center," she told AFP news agency.
"What if it shuts down? Many children will get worse, including my son."
Like MSF, the UN has warned it has been forced to divert resources from one community to aid another - most recently, from food programs to handle the escalating cholera outbreak.
The UN has said less than half of the $2.1bn pledged this year to Yemen by the international community has been disbursed.
But despite the dwindling hope, Yemenis refuse to give up on others in need.
"I am here to donate blood as an act of charity and for my country," said Abdullah Farei, one of a handful of citizens who turned up at the center to donate.
"We ask God that the citizens respond to the call of the blood bank and donate blood and money."