"If women don't take action, the authorities will not lift the ban. It is up to women to decide," Manal Alsharif, one of the campaign organizers who was detained last year after posting a video of her driving in the streets of the city of Khobar, told Reuters on Tuesday.
Under Saudi Arabia's strict Islamic laws, women require a male guardian's permission to travel abroad, undergo some types of medical surgery and work in some jobs.
While there is no written legislation banning women from driving, Saudi law requires citizens to use locally issued licenses while in the country. Such licenses are not issued to women, making it effectively illegal for them to drive.
"The key to lifting the ban on women driving in Saudi Arabia is to start with the women themselves ... That is why we ask the authorities to protect those women who need to practice that right," they said in an emailed statement sent to Reuters.
The campaign leaders had initially planned for women with valid international licenses to drive on June 17 but had to postpone their plans to June 29 after conservative Crown Prince Nayef passed away on Saturday.
"In commemoration of the one-year anniversary of the June 17 campaign, we renew our initiative by the women and men who support them in order to urge the authorities to look into this demand," they said in their original statement.
There is no indication of how many women will respond to this call and the Interior Ministry could not immediately comment.
"It is not likely that many women will take action on June 29. We are just stating a point that we are not giving up until the first (female) driver's license is issued," Alsharif said.
"We are hoping that with persistence and time ... people will see who we really are, Saudi women calling to have full citizenship and end decades of discrimination against us."
Around the same time last year, after pro-democracy protests swept through the region, dozens of Saudi women responded to the "Women 2 Drive" campaign, posting pictures and videos of their driving on twitter, Facebook and YouTube.
Some of the women were detained briefly and two faced charges, including that of "challenging the monarch". One of them was let go after signing a pledge not to drive again, while the other was sentenced to 10 lashes.
Reuters could not confirm if the lashing sentence was inflicted, but a Saudi princess tweeted that it had been revoked.
This time the "Women 2 Drive" activists are calling for King Abdullah to support them by stopping any punishments being imposed by the authorities.
King Abdullah has a reputation as a cautious reformer and supporter of women's rights. Last year he announced plans to allow women to vote in municipal council elections and join the consultative Shoura council.
"In our campaign we do not seek to disturb the authorities or violate rules and regulations ... All we want is for the women who need to go about their daily business and do not have a man to help her to be able to help herself," they said in a petition addressed to King Abdullah.
"Our only hope is hanging on your kindness and support for our campaign by instructing those involved from the police, regional governors, and the religious police to support those of us who have valid licenses to use them," they said.
(Reporting by Asma Alsharif; Editing by Angus McDowall and Andrew Roche)