Katea has already had two operations, one in Tonga and another in 2010 at Dunedin Hospital.
Katea's father Saia moved to Oamaru nearly five years ago.
Immigration New Zealand has told the family they must return to Tonga in February.
Immigration NZ general manager of visa services Nicola Hogg said the family was living in New Zealand unlawfully and was encouraged to return to Tonga voluntarily.
Mrs Tangifolau said claims the family have taken advantage of Katea's condition to remain here hurt deeply.
"No mum and dad would take advantage of their children," she said.
"We want her to have a better life and have better treatment."
Mr Tangifolau has been working at Lean Meats Oamaru Ltd for the past five seasons and manager Jeff Spillane said he has been a valuable employee.
"We would certainly welcome any legal means that would result in him being able to stay in the country and our workforce," he said.
The family are worried their daughter's foot will not be treated if they return home.
The operation in Tonga left Katea with a large scar and an infection in her foot, and Mrs Tangifolau was worried it would happen again if the procedure was not done in New Zealand.
"If it gets infected, she could lose her foot and I don't want that to happen," she said.
Mrs Tangifolau said the community had been donating time, support and money, which was overwhelming for the family.
"To all the people in Oamaru, we really appreciate it," she said.
"We feel like the community and the people are with us. They make us feel strong."