Sharing revenge-porn images and videos became a crime in England and Wales in February, but the law currently gives victims no right to hide identities.
The move was jointly started by Keeley Richards-Shaw, whose ex-boyfriend was the first sentenced under the new law.
The Ministry of Justice said judges had discretionary powers to withhold names.
Speaking on the BBC's Victoria Derbyshire programme earlier in December, Ms Richards-Shaw said media coverage had increased her distress and a change in the law was crucial to help victims "keep their personal life personal".
North Yorkshire Police and Crime Commissioner (PCC) Julia Mulligan, who is also behind the petition, has written to the government with Ms Richards-Shaw about the issue.
A loophole existed, the PCC said, because revenge porn was classed as a domestic abuse offence rather than a sexual offence.
She said: "It's wrong that victims of this very personal and distressing crime are being violated all over again by their stories being played out online and in the media.
"Any victim of revenge porn should have full confidence that their identity will be protected by law and this matter deserves to be far higher up the political agenda."

Keith Stewart found that the loch goes to a depth of 889ft (270.9m) on sonar equipment he uses.
The official maximum depth on the current chart for Loch Ness is 754ft (229.8m).
But the Maritime and Coastguard Agency (MCA), which is responsible for the official measurements, said it has no plans to resurvey the loch.
Mr Stewart's discovery has been dubbed a new hiding place for the loch's mythical monster, Nessie.
The MCA is responsible for mapping the sea and deep lochs and lakes around the UK.
It does this in order to maintain British Admiralty charts under the UK Civil Hydrography Programme.
An MCA spokeswoman said: "We currently have no plans to resurvey Loch Ness."

The victim suffered serious injuries - believed to be stab wounds - after he was attacked at a house in Auld Coal Grove, Bonnyrigg, at about 16:00 on Tuesday.
He was taken to the Royal Infirmary of Edinburgh, where he later died.
The arrested man is due to appear at Edinburgh Sheriff Court on Thursday.

The Conservatives want a report into the spending on the 218 "procurement cards" used for government spending.
A further £475,540 was spent in the first three months of 2015/16.
A review was conducted in April 2014, but Welsh ministers said they do not publish the results of internal audits.
However, all internal audits are shared with the public spending watchdog, the Wales Audit Office.
Welsh Conservative leader Andrew RT Davies said: "Unfortunately, the minister [Finance Minister Jane Hutt] is refusing to publish that report, which leaves a number of questions unanswered: is the taxpayer getting value for money?"
The Tories say they would encourage public bodies to publish all spending of more than £500.
A Welsh government spokesman said: "During the audit process for the Welsh government annual accounts, the Wales Audit Office has the opportunity to review all of the Welsh government's financial transactions.
"If Mr Davies is suggesting that civil servants are improperly using procurement cards then he should raise his concerns with the permanent secretary."
A petition calling for revenge porn victims to be given the same anonymity as other victims of sexual offences has gained more than 5,500 supporters.
[NEXT_CONCEPT]
A tour boat skipper has recorded a new deepest point in Loch Ness on his vessel's equipment.
[NEXT_CONCEPT]
A 36-year-old man has been charged with murder following the death of a 40-year-old man in Midlothian.
[NEXT_CONCEPT]
More than £3.8m has been spent in the last three years by civil servants using Welsh government-issued credit cards, prompting calls to reveal what the money has been spent on.