"Critical fuel shortages and a lack of medical supplies could force the Al-Sabeen Hospital to shut its doors within 48 hours," the humanitarian organization said late Sunday.
The hospital supported by Save the Children is the main facility for children and pregnant women in the area, and serves an estimated three million people, the organization said in a statement, AFP reported on Monday.
The Saudi-led coalition, which mounted an air campaign late March in support of exiled President Abed Rabbou Mansour Hadi, has imposed a blockade on areas controlled by Houthis.
The hospital was reliant on the Red Sea port of Hodeida for 90 percent of its imports, Save the Children said.
"The hospital has entirely run out of IV fluid, anaesthetic, blood transfusion tests, Valium to treat seizures and ready-prepared therapeutic food for severely malnourished children," the statement said citing the hospital's deputy manager Halel al-Bahri.
Fuel that the hospital acquired from the black market was enough to run power generators for two more days, he said.
Across Yemen, 15.2 million people are lacking access to basic healthcare, an increase of 40 percent since March, the organization warned.
More than half a million children are expected to suffer severe acute malnutrition this year, and there has been a 150 percent increase in hospital admissions for malnutrition since March, it said.
"It is crucial that enough medicines, supplies and fuel are able to get in to the country, otherwise the number of children dying from treatable illnesses is only going to get bigger," said Edward Santiago, Save the Children's Yemen director.