Authorities said a maximum "purple alert" was in effect from 9:00 am (0500 GMT) confining the entire population -- including law enforcement and emergency services -- to homes and offices.
"We expect a direct impact," said Celine Jauffret, an official of the Meteo France meteorological service.
In its latest bulletin, the agency said Cyclone Garance presents a "very significant" threat to the island.
La Reunion and nearby Mauritius -- around 225 kilometres (140 miles) to the northeast -- have been on high alert since Wednesday as Garance neared the popular tourist destinations.
Mauritius shut its main airport on Wednesday, while La Reunion did the same on Thursday.
"Faced with cyclone Garance, I call on our fellow citizens of Reunion Island to be extremely vigilant and to respect safety instructions," French President Emmanuel Macron posted on X late Thursday.
All the municipalities on La Reunion, home to around 900,000 people, have opened emergency shelters to the public, authorities said.
In the capital Saint-Denis, residents scrambled to stock up on essentials before the purple alert was issued.
"It took me 30 seconds to get my pack of water, and now I've been waiting at the checkout for 10 minutes," said Franck Vitry, waiting in line at a supermarket.
"I took rolls of puff pastry, eggs and sugar to make pastries with my children," said another customer, Maryvonne Laurent.
Inland, a vegetable farmer removed tarpaulins from his greenhouses with dread.
"It is a feeling of being powerless," said Jean-Christophe Hoareau.
Cyclone Belal, the last purple alert weather system, killed four people in January 2024.
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