The team, nicknamed Bafana Bafana, are due to face Jamaica in a friendly in Mexico on Friday.
They will then play in the World Cup opening match against Mexico, which is co-hosting the competition alongside the US and Canada, on 11 June.
“This Safa travel & visa debacle is embarrassing & grossly unfair towards the players & coaching staff,” McKenzie also wrote on X.
A brief statement from Safa said it was “working around the clock to ensure that the team travels to Mexico City as soon as possible ahead of the opening match”.
According to South African news site TimesLive, efforts were being made to get the team US visas to allow the team to travel either later on Sunday or on Monday.
The Bafana Bafana squad and staff need permission to enter the US as they play their second World Cup game against the Czech Republic in Atlanta.
South Africa is one of 10 African nations taking part in the expanded 48-team World Cup.
The game against Jamaica will be Bafana Bafana’s last chance to get ready for the tournament, which they last appeared at when South Africa hosted the finals in 2010.
Their final home match on Friday – a 0-0 draw against Nicaragua that included a missed South African penalty – was described in some reports as “disappointing” and extended their winless run to four games.