The small twin-engine plane – carrying Mr Missoni, 58, who owns the five-star Hotel Missoni in Edinburgh, his wife Maurizia Castiglioni, another couple and two Venezuelan crew members – went missing on Friday after taking off from Los Roques, an archipelago off the coast of Venezuela.
His sister Angela tweeted:"We believe Vittorio, Maurizia, Elda and Guido are still alive. Please help us find them and bring them back home."
Italy's Foreign Ministry declined to comment on how the search for the missing plane was progressing.
Italian media have suggested Mr Missoni, his wife and their friends Elda Scalvenzi and Guido Foresti may have been kidnapped.
The pilot of a Cessna 402 that took off one minute after the Britten Norman BN2 carrying Mr Missoni said he saw the plane disappear. pilot Enrique Rada said: "I saw them right in front of me as they were swallowed up by a huge cumulus cloud," said in a newspaper interview It must have been a lightning bolt."