Filed in the High Court's Sydney registry, the application asks the court to declare the law invalid or, alternatively, to issue injunctions preventing Ms Inman Grant and Ms Wells from enforcing the minimum age rules through compliance actions or notices.
As the deadline approaches, the Coalition has also clarified its position, after Opposition communications spokeswoman Melissa McIntosh suggested she may reassess the laws if implementation issues emerged.
On Thursday she reaffirmed support for the restrictions, saying: "The Coalition supports the social media ban. We all want this ban to work and to protect children," while noting concerns about additional platforms being brought under the law at short notice.
Ms Inman Grant said she did not expect platforms to seamlessly comply from day one.
"Some will do this really well and really precisely, and some will be a little bit slow and a little bit sloppy," she said.
The legislation requires companies to remove accounts belonging to users under 16 and take "reasonable" steps to prevent underage access. Although platforms risk penalties of up to $49.5 million for failing to comply, the commissioner said enforcement would begin with a "gradual and fair" process before fines were issued.
Platforms will soon be required to provide evidence that they are deactivating accounts belonging to Australian teenagers, with eSafety expecting weekly reports initially, moving to monthly updates.
"We will be compelling them to provide very specific information about what they've done and how many accounts they've deactivated or de-platformed," she said.