The petition calls for the kingdom's women to be treated "as a full citizen, and decide an age where she will be an adult and will be responsible for her own acts", said campaigner Aziza Al-Yousef of Riyadh.
The retired university professor told AFP that she tried unsuccessfully to deliver the petition with 14,700 names to the Royal Court yesterday.
The activists will now send it by mail as requested.
Under the guardianship system a male family member, normally the father, husband or brother, must grant permission for a woman's study, travel and other activities.
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Activists say that even female prisoners have to be received by the guardian upon their release, meaning that some have to languish in jail beyond their sentences if the man does not want to accept them.
"We are suffering from this guardianship system," said Nassima al-Sadah, an activist in Eastern Province.
The campaign is an outgrowth of a Twitter hashtag in Arabic that started more than two months ago calling for an end to guardianship.
"Saudi Arabia's male guardianship system remains the most significant impediment to women's rights in the country despite limited reforms over the last decade," the watchdog said.
Activists said that if they have open-minded male family members, getting their consent is not a problem - although the men still have to formally sign consent papers.
"It's a government system" which only came into effect about 30 years ago, Sadah said.
Activists said that ending guardianship will make it easier for women to work at a time when the kingdom wants to boost female employment.
In April, Deputy Crown Prince Mohammed bin Salman announced the wide-ranging Vision 2030 plan to diversify the oil-dependent economy.
Under the National Transformation Programme (NTP), which sets targets for implementing the Vision, the proportion of women in the workforce should rise from 23 to 28 percent by 2020.