National leader Simon Bridges says Finance Minister Grant Robertson and Treasury Secretary Gabriel Makhlouf must resign.
And he has called on Deputy Prime Minister Winston Peters, who yesterday said that National had acted illegally in obtaining Budget 2019 information, to publicly apologise.
"It shows deep dishonesty. Treasury has known since Tuesday exactly what happened and they covered it up to hide their incompetence. They have sat on a lie, calling the National Party criminal hackers and calling in the police," Bridges told reporters this morning.
Bridges said Robertson was "donkey deep" in this.
"He does not have the moral authority to deliver the Government's Budget today."
Early this morning, Treasury revealed that sensitive Budget 2019 information that was obtained from its website was not hacked, but instead accessed legally.
Makhlouf had earlier claimed the website had been hacked and had referred the matter to police.
However, police said today the person or persons were able to "exploit" the system because Treasury staff had been preparing a clone website in the background that they intended to swap over with the live website on Budget day.
Bridges said that National staffers had "stumbled upon" Budget 2019 information by doing simple searches on Treasury's website, something that New Zealanders do every day on Google or Trade Me.
"Any member of the NZ public could have done this if they had an interest in the Budget, from your grandson to your grandma."
Bridges said that people in Treasury had told him that Treasury was aware of what had happened on Tuesday, but had continued to "lie" and called in the police anyway.
Makhlouf had likened the "hack" to someone attacking a padlock on a door, but Bridges said it was more like putting information out on to the street with a "free to a good home" sign on it.
Bridges previously said that Robertson had implied that National had hacked the Treasury. He accused Robertson of lying and unfairly smearing National, and this morning repeated his call for Robertson to resign.
Robertson has said he will not resign, and responded this morning: "I'm very disappointed that confidential Budget information was able to be accessed in this way. I am also very disappointed that the Treasury did not seek to find more information as to how this happened before referring the matter to the Police.
"I now await the inquiry of the State Services Commissioner into this matter."
Robertson earlier this week said he had contacted National, asking Bridges not to release any more information – "given that the Treasury said they have sufficient evidence that indicates the material is a result of a systematic hack and is now subject to a police investigation".
Robertson and Prime Minister Jacinda Ardern have repeatedly tried to distance the Government from any implication that National had hacked the Treasury website.
But Peters said yesterday that he knew that National had acted illegally.
Bridges said the only wrongful behaviour came from Treasury, Robertson and Peters.
"The Deputy PM of New Zealand came out and accused me of criminal behaviour. That is disgraceful," Bridges said.
He said if Peters had any integrity, he should publicly apologise.
He said Peters' comments prompted him to reveal this morning how National had come upon the information.
The public interest in National releasing Budget information was to expose the Government's incompetence.
Early this morning, Makhlouf said: "In my view, there were deliberate, exhaustive and sustained attempts to gain unauthorised access to embargoed data. Our systems were clearly susceptible to such unacceptable behaviour, in breach of the long-standing convention around Budget confidentiality, and we will undertake a review to make them more robust."
Makhlouf did not offer to resign, though he has already leaving on June 27 to become the equivalent of Reserve Bank Governor in Ireland.
The Treasury website was exploited because staff had begun uploading some Budget information on to the clone site.
Although not publicly accessible, some information could be seen when a search was made on the website.
"The evidence shows deliberate, systematic and persistent searching of a website that was clearly not intended to be public," Treasury said.
The news comes after speculation had begun to mount that the Budget leak was more "cock-up" than hack.
NZ Herald tech writer Juha Saarinen found screenshots of a Google search for "estimates of appropriation 2019/2020" circulating on Twitter suggesting the data was publicly accessible.
An inquiry into the strength of security around the Budget will now be undertaken by the State Services Commission.