Postal vote forms for the plebiscite are meant to be sent next month, with the final result being announced on November 15.
Lawyers acting on behalf of independent MP Andrew Wilkie, Parents and Friends of Lesbians and Gays (PFLAG) and lesbian parent Felicity Marlowe explained their case to reporters shortly after lodging the challenge.
"We're trying to stop the postal vote going ahead because we believe it's not within the power of government to do so unless parliament authorises it," Public Interest Advocacy Centre CEO Jonathon Hunyor said.
"This is uncharted waters, this is going to be a challenging case."
Mr Hunyor said his case would rest upon several arguments.
"Firstly, we'll be arguing the finance minister lacks the power to approve the funding in this case because there is no real urgency," he said.
"Secondly, the legislation that gives the finance minister the power to allocate these sorts of funds without parliamentary authorisation is itself invalid."
He will also argue the Australian Bureau of Statistics is the wrong agency to carry out the non-binding public ballot.
"The exercise simply isn't a statistical one and so it doesn't fall in the power of the ABS," Mr Hunyor said.
"We're going to be arguing the government can not validly undertake a postal vote and also cannot fund the exercise if it doesn't have the authorisation of power."
Earlier this week, Finance Minister Mathias Cormann, who is also Acting Special Minister of State, said the government was confident it had "a legal and constitutional way forward to give the Australian people a say through a non-legislated postal plebiscite".
The senate voted down the government's second attempt at a national plebiscite, forcing it to go to its second option of the postal ballot.
Opposition leader Bill Shorten committed Labor to mounting a 'yes' case for the plebiscite in parliament on Thursday.
"It will vilify and demean LGBTI Australians and their children," he said of the postal vote.
"I will be voting yes. I will be campaigning for a yes vote. No true leader is every too busy to fight for the fair go in this country."
And he personally singled out Prime Minister Turnbull for criticism.
"I hold you responsible for every hurtful bit of filth that this debate will unleash," he said.