The St John team said it was transporting one male, aged approximately in his late 30s, to Dunedin Hospital with "very serious injuries".
A Worksafe New Zealand spokeswoman said the organisation had been notified and was investigating.
It was too early to say when the investigation would be wrapped up, she said.
Port Otago general manager infrastructure Lincoln Coe refused to comment.
He said the port would make a comment generally available when it was able to do so.
The Maritime Union is calling for an industry-wide health and safety inquiry into all dockwork operations, in light of the incident.
The driver who was seriously injured in the crash was a member of the Maritime Union.
Maritime Union of New Zealand national secretary Joe Fleetwood said the union would be following the Worksafe NZ investigation closely and supporting the member and his family.
Ongoing accidents in ports throughout New Zealand were a concern, he said.
The latest incident reinforces the union's call for an industry-wide health and safety inquiry into all stevedoring - or wharf manual labouring - operations in New Zealand, he said.