The 1943 Harvard was taking off towards the west when it veered off the runway on to the grass where its undercarriage collapsed.
It ended up sitting on its fuselage on grass next to the main runway.
The two pilots on board walked away uninjured, police said.
Emergency services, initially concerned about fire, have now left the scene.
Warbirds Over Wanaka International Airshow general manager Ed Taylor told the Otago Daily times he did not know what went wrong but was grateful no one was injured.
Some light aircraft have been able to take off on a shortened runway but there are still several others unable to leave.
Attempts are being made to move the crashed plane away from the runway using a truck with a crane hoist.
The ODT has received reports of a second Harvard involved in an incident earlier this morning.
It is understood the Harvard abandoned its take off after one wing tip touched the tarmac.