Witnesses reported seeing thick smoke as the blaze "completely destroyed" the shuttle bus, parked in a waiting area.
The fire also caused smoke damage to "much of the front of the terminal building," firefighters said.
Bosses at the Essex airport apologised and advised passengers to re-book their flights with their airlines.
On Friday evening, the airport said no injuries had been reported and that the fire - which was extinguished at about 17:00 BST - had been caused by an engine electrical fault on the bus.
"The fire has since been put out and we're working on returning to normal operations," a spokesperson added.
Later, however, the airport tweeted to announce the cancellation of all outgoing flights.
Essex Fire Service said four fire crews had been sent to the scene.
