Their attacks have driven Nigerian oil output to near a 22-year low and, if the violence escalates into another insurgency in the restive area, it could cripple production in a country facing a growing economic crisis.
President Muhammadu Buhari has said he will crush the militants, but a wide-scale conflict could stretch security forces already battling a northern rebellion by hardline Sunni Muslim group Boko Haram.
Militancy has been rife over the past decade in the Delta, a southern region which is one of the country's poorest areas despite generating 70 percent of state income.
Violence has increased sharply this year - most of it claimed by the "Avengers" - after Buhari scaled back an amnesty deal with rebel groups, which had ended a 2004-2009 insurgency.
Under the deal, more state cash was channelled to the region for job training and militant groups were handed contracts to protect the pipelines they once bombed. But Buhari cut the budget allocated to the plan by about 70 percent and cancelled the contracts, citing corruption and mismanagement of funds.
The "Avengers" have carried out a string of attacks since February that reduced Nigerian oil output by at least 300,000 barrels a day of output, and shut down two refineries and a major export terminal.
On Thursday the group emailed journalists a statement saying it was fighting for an independent Delta and would step up its attacks unless oil firms left the region within two weeks.
"If at the end of the ultimatum you are still operating, we will blow up all the locations," it said. "It will be bloody. So just shut down your operations and leave."
"To international oil companies, this is just the beginning and you have not seen anything yet. We will make you suffer," it said.
Authorities have no hard facts about the group - such as its size, bases or leadership, Nigeria-based diplomats say.
Diplomats and security experts say it has shown a level of sophistication not seen since the peak of the 2004-2009 insurgency, which halved Nigeria's oil output. They say it must be getting help from sympathetic oil workers in identifying the pipelines to cause maximum damage.
"Its scary. Their demands are impossible to meet so there will be probably more attacks," said a security expert, asking not to be named.
In February the group claimed an attack on an undersea pipeline that forced Shell to shut a 250,000 barrels a day Forcados terminal. Last week, it took credit for blasting a Chevron platform, shutting the Warri and Kaduna refineries. Power outages across Nigeria worsened as gas supplies were also affected.
There have been other smaller attacks and this week another explosion, which bore the hallmarks of the group, closed Shell's Bonny Light export programme.
Reuters, like other media, has been unable to reach the group, which mainly communicates via Twitter, with the location tracker switched off, and on its website.
Its members describe themselves there as "young, well travelled" and mostly educated in eastern Europe.
Given the lack of intelligence about the militants, the army launched a wide-ranging hunt across the Delta this week, sending gunboats into mosquito-infested creeks and searching villages in the middle of the night.
But some residents say such a heavy-handed military approach stokes dissent in the Delta where many complain of poverty despite sitting on much of Nigeria's energy wealth. They say some villagers help militants to hide in the hard-to-access swamps.
"The military came at 12.30 am with two gunboats ... they went from house to house. Many ran into the bush," said Godspower Gbenekemam, chief of the Gbaramatu area.
"The military stayed on until about 5.30 am, during which nobody was able to move out," he said. "We are not part of the people blowing up pipelines. We do not know them so the military should leave our community alone."
Alagoa Morris, an environmental activist based in the Delta, said unless soldiers acted with restraint, more people would join the militants, with a risk of "the Niger Delta returning to another round of full-scale militancy".
Even oil majors, which have long pressed for better pipeline protection, worry the tactics could backfire.
Executives met Vice President Yemi Osinbajo this week and one of them warned the government was being "too direct and blunt" and needed to find some balance, according to a source familiar with the discussions.
The military has not said how many soldiers have been involved in the sweep. The army searched several villages around Gbaramatu because that part of the Delta is home to former militant leader Government Ekpemupolo, better known as Tompolo.
Some officials have linked Tompolo to the "Avengers", pointing to the fact that the attacks began after authorities issued an arrest warrant for Tompolo on graft charges in January.
Tompolo has denied any ties, saying he himself is a victim as the group had asked him to apologise for criticising it.
For Buhari, the campaign against former militants is a part of his election promise to fix a country gripped by graft and mismanagement, but many locals in the Christian south see him, a Muslim northerner, as an oppressor.
Buhari's cutting of the amnesty plan's budget has also caused widespread resentment in the Delta, as it helps fund job training for the unemployed.
Tapping into such anger, the "Avengers" point out that the former military ruler has never visited the Delta, where many roads are pot-holed and some villages are polluted from oil spills.
In a flurry of statements, the militants have published a list of demands, from cleaning up oil spills to keeping the amnesty plan, leading up to Thursday's ultimatum.
Diplomats say some of Tompolo's followers have probably joined the "Avengers" and that the group's ranks could be swelled by an army of unemployed willing to work for anyone.
But, adding to the confusion surrounding the group, some former rebels have denied knowledge of the militants and say they have brought unwanted military attention to the area.
"Niger Delta Avengers are not fighting for the sake of Niger Delta," said Eris Paul, a former leader of the now-defunct Movement for the Emancipation of the Niger Delta (MEND), which was one of the most powerful militant groups. "We don't know them." (Additional reporting by Anamesere Igboeroteonwu, Ron Bousso and Libbby George; Editing by Pravin Char)