You are not permitted to download, save or email this image. Visit image gallery to purchase the image.
The Grounded Kiwis petition was launched earlier this week and has received more than 3000 signatures so far.
The group say that New Zealanders abroad feel they have been abandoned and that the MIQ system is broken.
Alexandra Birt, a Kiwi living in London who drafted the petition, told RNZ's Morning Report programme today she wanted to return home to see family, but many others had their own reasons.
"New Zealanders at home and abroad are really affected by the system at the moment, and a general essence of it for those of us abroad, is that we can't get home.
"And for those who are in New Zealand who have urgent needs to go overseas, they can't do that at the moment because you just simply can't get space in MIQ. So that's why I set up the petition."
She said there needed to be a fairer booking system.
"That looks like a system that's transparent, accessible, minimises uncertainty, and ultimately just one that allows people to get home.
"It's a system that can't be subverted by bots and scripts or paying third parties which we've heard a lot about in the media recently; one that meets the accessibility standards, so everyone has an equal opportunity to book.
"There's been a lot of issues with emergency allocations where people have extremely valid emergency circumstances, but they just can't obtain emergency allocation and ultimately what it all comes down towards one where supply meets demand."
She said the system needed to be upfront about wait times.
The Government and MBIE also needed to look at the science to allow travel options for those who had been vaccinated, Birt said.
"One of the complaints was ... on the basis that the current system, as it's designed, is inaccessible and discriminates against particularly older disabled and blind people who can't complete the steps quickly enough to secure a slot for people that are relying on screen readers who just are not in the same position as other people might be who are able to utilise these kinds of high-tech speedy solutions."
The Human Rights Commission has received 30 complaints about the online booking portal, including claims that it discriminates against older people, disabled people and blind people.
"I feel like the government just doesn't get it."
Birt said the Government "kept referring to the situation as being analogous to a sporting event, and to me that just boggles my mind when people are trying to get home".
"I had a friend whose mum died overnight and she couldn't get home to New Zealand because once someone has died, we no longer meet the emergency criteria to get back.
"So she was separated from her family in that awful situation, and to analogise that with trying to get a ticket to a sporting event, it's just ... they don't get it, and that's why we're campaigning for change. It's both changing the perception and actually seeking action."
Last month, MIQ deputy chief executive Megan Main said a fairer system was in the works but "the reality is some people will miss out".