Genevieve Cheatham's petition calling for Joyce to resign or Prime Minister Malcolm Turnbull to push for his resignation had received 6900 signatures on change.org as of Wednesday morning.
"Barnaby Joyce must resign. We do not want him to represent us as Deputy Prime Minister of Australia," Ms Cheatham posted.
Mr Joyce said yesterday that he did not break any parliamentary rules while conducting an extramarital affair with a former staffer.
The Catholic, who has campaigned on "family values" and has been married for 24 years, is expecting a child with his former press secretary Vikki Campion.
He is under immense pressure for failing to declare his relationship with his former staffer when she was given two highly-paid positions working for the ruling coalition government.
Australia's ministerial code of conduct requires lawmakers to declare relationships with any staff member of a cabinet minister. Joyce said he did not break any rules because Ms Campion was not his "partner" at the time.
The scandal could have major repercussions for Australia's centre-right government. If Joyce does resign from parliament, Prime Minister Malcolm Turnbull will lose his razor-thin one seat majority in parliament.
Mr Turnbull has not asked his Liberal deputy to ready herself to be acting prime minister while he's in the United States next week in case Mr Joyce is not available to do the job.
If both the prime minister and deputy prime minister are unable to take Australia's reigns, Foreign Minister Julie Bishop would ordinarily be next in line.
There is pressure on the Mr Joyce, the Nationals leader, to step aside or at least take leave rather than act in the top job amid ongoing scrutiny of his actions.
But Ms Bishop, who has plans to be overseas herself next week, said Mr Turnbull had not asked her to be available.
"I'm over in Kuwait (this week) with back-to-back meetings from very early in the morning to late at night and focusing on Australia's military intervention and humanitarian contribution to Iraq so it's not a matter that I'm focusing on," she told ABC radio on Wednesday.
"I am returning to Australia from Kuwait; I do have plans to be overseas next week - parliament is not sitting - if circumstances change then, of course, I would change plans."
However, she said her understanding was the usual arrangements would apply next week.
