Officials said seven boats capsized and three cars were washed away by floods as Cyclone Mekunu slammed into the island, which lies between the Arabian Peninsula and the Horn of Africa.
Most parts of the island have lost communications while meteorologists said the storm is worsening and is now headed for Oman.
The same weather system has already hit the Horn of Africa, killing more than 50 people in Somaliland on Wednesday.
Evacuations to shelters are underway in Socatra and 200 families have been evacuated from their villages, officials said.
The Saudi ambassador Mohammed al-Jaber said tens of thousands of tons of relief, shelter and medical items are being prepared to be transported to the island, provided by the King Salman Humanitarian Aid and Relief Centre.
The storm, which has now intensified to Category 2, is expected to reach southern Yemen and the coast of neighbouring Oman, Oman's state news agency reported.
Yemen is already grappling with one of the world's worst humanitarian crises as a result of conflict.
The three-year war has killed more than 10,000, displaced three million people, triggered a cholera outbreak and pushed the impoverished country to the verge of starvation, according to the United Nations.
Additional reporting by Reuters
