Police were called by the ambulance service shortly before 01:40 BST after the discovery at Waterglade Industrial Park in Eastern Avenue, Grays.
The lorry driver, a 25-year-old man from Northern Ireland, has been arrested on suspicion of murder.
Essex Police said the lorry travelled from Bulgaria and entered the country via Holyhead, Anglesey, on Saturday.
Early indications are that 38 adults and one teenager have died, the force said.
Ch Supt Andrew Mariner said officers were trying to identify victims but anticipated a “lengthy process”.
“We have arrested the lorry driver in connection with the incident, who remains in police custody as our inquiries continue,” he said.
A cordon has been put in place and the Waterglade Industrial Park is closed off.