Photos and videos posted online showed water flowing through the streets of the provincial capital, Hadibu.
It is believed to be the most powerful storm that Yemen has seen in decades.
The UN`s World Meteorological Organisation described the cyclone as "extremely severe", and said that sea conditions around the centre of the storm were "phenomenal".
At 06:00 GMT on Monday, the US Navy`s Joint Typhoon Warning Centre reported that Chapala was generating gusts of up to 240km/h (150mph), equivalent to a category 4 hurricane.
Gusts of up to 165 km/h are forecast for when the storm makes landfall just west of the city of Mukalla, on the south coast of the Yemeni mainland, at around 06:00 GMT on Tuesday.
The cyclone could deluge parts of the country with up to 500mm (20in) of rain in two days - 10 times the annual average.
Socotra is situated 368km (230 miles) south of the coast of Yemen in the Arabian Sea, to the east of Somalia.
It is home to about 50,000 people, who speak their own language, and hundreds of exotic plant species found nowhere else on earth, including dragon`s blood trees.
The mayor of Hadibu, Salem Zaher, told the AFP news agency that Chapala had damaged more than 80 houses and left hundreds of people needing hospital treatment.
More than 1,000 families had been evacuated and resettled in schools and camps inland before the storm hit, he added.
Residents of Mukalla, which has been controlled by a tribal council and jihadist militants from al-Qaeda in the Arabian Peninsula (AQAP) since April, meanwhile expressed concern about local preparations for when Chapala made landfall.
"The sea water level has risen by 9m (29ft) and has destroyed the Mukalla seafront," resident Mohammed Ba Zuhair told the Reuters news agency.
"Many people have left their homes and are seeking refuge in schools. No relief or aid efforts are under way by either the tribal council or al-Qaeda, and the situation is really bad."
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