The team had been due to leave on a charter flight for their training base in the Mexican city of Pachuca on Sunday but that was postponed while the South African Football Association (SAFA) tried to sort out the paperwork.
"All Bafana Bafana players received their visas to travel to the USA. Outstanding is assistant coach, team doctor, head of security and one analyst. The charter will leave tomorrow (Monday)," McKenzie said on X.
He had earlier issued a furious rebuke to SAFA over the paperwork issues, calling them "embarrassing and grossly unfair towards the players and coaching staff", adding "we are being made to look like fools".
The football association confirmed in a statement that the team had been unable to fly in the morning as planned due to "challenges regarding visas" for some players and officials.
SAFA has not yet confirmed the team's departure with an emergency meeting planned later on Sunday to discuss the issue.
South Africa kick off the World Cup against Mexico on June 11, and are appearing at the global finals for the first time since they were hosts themselves in 2010.
Coach Hugo Broos had previously spoken of wanting to be in Mexico by June 1 at the latest to allow his players to recover from jetlag and acclimatise to the altitude, though they will now arrive a day later.
Pachuca is around 680 metres (2,200 feet) further above sea level than Johannesburg at 2,432m.
South Africa are due to play Jamaica in Mexico on Friday in a final warm-up fixture.
An administrative bungle in qualifying almost cost them their place at the finals when their 2-0 victory over Lesotho was overturned after they fielded an ineligible player in Teboho Mokoena, who was suspended for the clash.
Despite this, they still finished a point ahead of Nigeria and Benin in their qualification pool.
South Africa also have Group A games against Czech Republic (June 18) in the U.S. city of Atlanta and South Korea (June 24) in Monterrey back in Mexico.