At issue are emails to or from Bob Nazarian which the commission believes are directly related to the Algo Centre Mall.
As a required prelude to court action, Commissioner Paul Belanger has already formally ordered Nazarian to comply with several summonses to turn over the material.
Nazarian is a key figure in light of allegations — strenuously denied — that he failed to carry out proper maintenance, which might have averted last June's tragedy that killed two women.
Belanger's order, issued last month but which has had no response to date, also applies to Nazarian's wife Irene and son Levon.
The legal battle, which commission counsel refuse to discuss publicly, has seen a series of futile attempts — dating to September — for production of the documents.
"Email and webmail accounts had been used to conduct matters relating to the Algo Centre Mall by Robert, Levon and Irene Nazarian," Belanger's order states.
"No emails from these accounts had been produced."
At one point, the family's lawyers claimed provincial police had seized the material as part of a criminal probe into the mall's collapse.
Other requests were met with promises of producing the materials right away, or simply weren't acknowledged at all.
Sources said the emails in question were on American servers and provincial police had not found them.
Nazarian could not be reached at home Tuesday, but when asked about possible legal action to force production of the documents, one of his lawyers insisted they were working diligently to gather and screen the requested material.
"It's not an attempt to bury documents or hold anything back," lawyer Michael Title said from Toronto.
The summonses involve about 40,000 emails, some of which Title called "private and utterly irrelevant."
Captured by the commission's requests were also Levon Nazarian's emails.
"There was some resistance to the intrusiveness of this," Title said.
"(Levon Nazarian) was not terribly happy with the request (and) it took some prodding to have him comply."
Hearings into the tragedy began this week and are proceeding chronologically, starting with the design and construction of the mall.
Nazarian is expected to testify in about four or five weeks time and the commission was hoping to have the emails in hand long before then.
Title said they would comply with Belanger's order but did not say when.
"It's a large task to screen the relevant from the irrelevant," he said. "We only have 4,000 emails left."
Still, sources familiar with the situation said the commission was fed up with the delays and would be looking to Ontario's Divisional Court to force compliance.
"The hammer is about to come down," one source said.
On June 23, 2012, part of the mall's roof-top parking deck, which had serious water-penetration issues since it was built in 1982, collapsed into the stores below.
Investigators believe that a weld between a steel I-beam and an upright column failed because of years of rust and salt corrosion.
One issue for the inquiry is why numerous inspections apparently failed to pick up on just how serious the problem was.
On Tuesday, the inquiry heard from Dale Craig, a professional engineer, who provided an overview of putting up a large building, from design to completion.
"Maintenance is often ignored until problems occur," he said. "Out of sight, out of mind."
A structural engineer involved with the building of the mall and who apparently lost his licence over problems with several other building projects, is slated to testify on Wednesday.