But Delahunty said dissatisfaction with Shaw may stem from his focus on consensus within parliament in his role as Climate Change Minister.
Instead, the Greens should be the strongest and most vocal party on climate change, and call the Labour party to account, she said.
"What he has believed to be the best strategy is not necessarily supported by everybody because it's not resulting in pressuring Labour to take stronger action - in fact, it's seen as very weak by many of us involved in activism, and I think obviously by some of the party members as well."
Delahunty said it was important to focus on the vision other potential leaders offered, rather than speculating on which individual might replace Shaw.
She said the Greens needed transformative policies to stop climate change, defend the vulnerable in society, and improve social justice.
"I can't encapsulate that and say there's one individual right now that I would support," she said.
"It's more about if James doesn't stand again, it opens up an opportunity for somebody to come out and show some vision, and we really need to hear that. It's got to be different from a middle-class, middle-age party just propping up the Labour government."
Not easy being Green: Shaw to seek party feedback
Shaw said he wanted to sound out the party's membership before confirming whether he would run for the position again or not.
He indicated he was inclined to put his hand up again, but needed to find out if that was what members wanted first.
"The question that I've got is if the nominations close and I'm the sole nominee again, will the vote go exactly the same way or will it change? And so that's really my one hesitation," he said.
Green Party co-leader Marama Davidson, whose position was reconfirmed by the delegates votes, wouldn't say concretely whether she wanted Shaw to be her co-leader again.
However, she said he had worked hard in the role.
Davidson said she was shocked and saddened, but wouldn't get ahead of the party's nominations by saying whether she'd back Shaw for the role.
"I simply cannot pre-empt that conversation but what I can do without damaging the process of the Green Party is be really clear about how he's slogged his guts out behind the scenes."
Nominations for Shaw's spot would be open for a week, with any election to be held in a month.
- RNZ
