The crib, which is 75sq m, has not changed hands since it was built 87 years ago, and has come to the end of its 30-year concession from the Department of Conservation.
The family says it is "profoundly concerned" about Doc’s decision not to continue the lease on the crib.
In a letter to Doc, applicant Michelle Matthews said the crib had served as a vital link to their ancestors’ history.
"Our family has always respected Doc’s values of stewardship, partnership and respect for nature, but it is incredibly difficult for us to comprehend how removing this crib, which connects us to our heritage and the land, aligns with these principles."
The crib has been in the same place since 1938, is not used as a commercial operation, is not accessible by road and is too brittle to be moved.
In an application to Doc, the family said it had deep roots in the Glenorchy area.
The crib is shared by family members who are third-, fourth- and fifth-generation descendants of Roy and Helen Elliott.
They are Mrs Matthews, Carol Evans, Robin Evans, Rhonda Dunbavan and Clifton Todd.
The applicants are also related to Tommy Thomson, who served on the Lakes County Council for 33 years, farmed Mt Earnslaw Station and was a major force behind the establishment of the road between Queenstown and Glenorchy.
Mr Elliot farmed Routeburn Station with his wife’s brothers in the early 1900s.
He built the crib in 1938 on land which had grazing rights held by Woodbine Station. The station was owned by Huntley Groves, the brother of Mrs Elliot. Rates were paid to the councils of the time until 1995, when it was changed to a new agreement with Doc.
In its application, the family said they had maintained the property in its original condition and had kept the area around the crib clean and tidy.
"We have followed all restrictions on the lease, such as those preventing us from using the crib full-time for accommodation.
"The family has continued to look after the area immediately around the crib and this includes collecting rubbish and waste left by others using the beach."
The crib is often used for wedding photos and tour visits.
Hollywood star Brad Pitt used the crib when he was filming the movie Heart of the Beast in the area earlier this year.
Mrs Matthews said, in her letter, the crib played an integral role in ensuring family memories lived on.
"It is heartbreaking to think that without this crib, these stories might not be preserved in the same way. It is not just a structure to us, but a living connection to the past that has been passed down through generations.
"The crib is a sanctuary for us, one that is deeply woven into our family’s fabric, and it has provided a space for our memories and traditions for generations."
Doc Wakatipu wai Māori operations manager David Butt said the 30-year term expired in June.
"We’ve received an application for a lease for a further 30-year term, which has been publicly notified. Any private or commercial facility on public conservation land requires a concession."
Once submissions close, Doc would consider all submissions, and a hearing might be held as a result. No decisions had yet been made, he said.
