The officials said Cyclone Chapala seriously damaged or destroyed at least 20 homes on the island Sunday, where trees have been uprooted and fishing boats sank.
They said many residents living near the coast are taking shelter in caves or in schools, The Associated Press reported.
The island is being hit by heavy rain and strong winds, and rising waves are battering the coast. The officials said coastal areas are flooded.
Authorities on the island have transported 9,000 people from their homes to other places to take refuge, mainly schools, said Socotra Deputy Governor Ramzi Mahfouz. But the evacuation was only carried out in the provincial capital, Hadibo, and residents elsewhere are facing more challenges. The situation is being compounded by a shortage of fuel on the island for more than 20 days, he said.
Mohammed Salmein, a resident of the island, said they haven't seen winds and rain like this in decades.
"We are expecting a true disaster to befall us," he said, "and only God knows whether we will last until tomorrow."
Socotra island lies in the Indian Ocean some 230 miles (368 kilometers) off the coast of Yemen's mainland. It is known for its unique vegetation and ecosystems, including rare dragon blood trees.
The UN weather agency said on Friday that Cyclone Chapala could dump more than a year's worth of rainfall on the arid, impoverished and war-torn country.
The Global Disaster Alert and Coordination System forecasts the storm will make landfall in Yemen around midday on Tuesday, following earlier estimates that it would hit on Monday.
It said that up to 390,000 people could be affected by the wind speeds, and 3,500 people residing in coastal areas could be affected by storm surge.
In Al Mahra, Yemen's easternmost province, security officials said dark clouds are forming amid unusually strong winds, though no damage has been reported yet.
Neighbouring province Hadramawt, which is partially controlled by Al Qaeda, is also bracing for the storm. The Emergency Council in Hadramawt appealed to citizens living in coastal areas to evacuate their homes and move to high ground.
Mukalla, a city under Al Qaeda control, is evacuating a coastal neighbourhood which is one of the oldest in the city.