The party's sports and recreation spokesperson Solly Malatsi said in a statement that the application would be submitted to the SA Football Association and the sports and recreation department.
"Safa president Danny Jordaan, and former president Molefi Oliphant, have so far failed to provide the nation with credible information refuting the bribery allegations," he said.
"The DA believes that, at all times, the government must act honestly, transparently and in the best interests of all South Africans. While we will never forget the euphoria of the World Cup, it is only fair for the implicated officials to account to the South African people."
On Tuesday, Parliament's portfolio committee on sport decided there was no reason to summon Jordaan to come and explain the bribery scandal.
Jordaan and Oliphant had reportedly signed letters giving instruction that the $10m be channelled to accounts managed by deposed Fifa vice president Warner.
US authorities allege that this payment was a bribe to help secure the World Cup bid for South Africa.
Jordaan, recently appointed mayor of Nelson Mandela Bay, and Oliphant are yet to break their silence on the bribery allegations.
Malatsi said it was unfortunate that a PAIA had to be submitted for the documents, since the ANC was shielding Jordaan from accounting to Parliament.
"It is clear that those intimately involved in the bid process and subsequent payment are not willing to be open and honest," he said.
"What government or organisation would authorise the payment of $10m without documentation or regular reporting on how the funds were spent?"
It was telling that neither Safa, nor the South African government, had attempted to hold the Confederation of North, Central American and Caribbean Association Football (Concacaf) accountable, he said.