The referendums are not initiated often and are seldom successful.
The last was in 2012, initiated by Roy Reid and asking people whether they supported the sell-off of state owned assets.
Despite this, Tricia Cheel of Friends of Sherwood, was optimistic the group would gather enough signatures.
"When people realise the extent of the suffering that's going on ... I don't think we're going to have trouble even getting 400,000 signatures," she said.
"We simply cannot condone this level of cruelty under any circumstances."
The Friends of Sherwood went to the Environment Court in September and won a court injunction temporarily suspending a plan by Auckland Council to drop 1080 poison in the Hunua Ranges.
But in a later judgment, the court allowed it to go ahead, saying it was not persuaded that there was likely to be serious harm to the environment if the drop went ahead.