The family said on the page they were "still hopeful" he would be found alive.
"Reports have emerged from the found survivors that all but three crew members were on the ship's bridge prior to the boat sinking and were preparing to board the life vessels. All life vessels hold 20-30 people and include enough food and water rations for 30 days," they wrote.
"Throughout this, we believe the New Zealand Government has failed to act to resume the search which we believe was prematurely called off."
The funds raised by the page will be pooled with funds raised by Australians involved in the search efforts.
"The private search efforts could involve satellite and drone surveillance technology, chartered search-and-rescue devices, media awareness campaigns and search incentives," they wrote.
"People have survived 41 days on a life raft in this very area before.
"Lochie is a legend, loved by many. He is extremely capable, adventurous and without a doubt possesses the character needed to not only get through this but to help pull others through."
The Transport Accident Investigation Commission (TAIC) has announced it will assist Panama in the investigation by collecting information from New Zealand organisations.
"Under international protocols, investigations into accidents on the high seas fall to the vessel's flag state – in this case, Panama," TAIC said in a statement.
"TAIC's legislation does not allow the commission to open its own inquiry, although it can co-operate and co-ordinate with other accident investigation organisations overseas, including taking evidence on their behalf.
"Consequently, TAIC asked, under the International Maritime Organisation's casualty code, to participate in Panama's investigation as a 'substantially interested state'.
"A state is 'substantially interested' if, as a result of the marine casualty, nationals of that state lose their lives, or has important information at its disposal that the marine safety investigating state considers useful to the investigation."
Following the sinking, animal rights group SAFE and the SPCA have urged the Government to ban live exports of animals.
"The tragic sinking of Gulf Livestock 1 is yet another live export disaster that has put this cruel trade back on the agenda both internationally and for Kiwis at home," said SAFE chief executive Debra Ashton.
"Live export is less than 1 per cent of our overall exports but it has a big impact on our reputation, putting other exports at risk."
Ashton and SPCA chief executive Andrea Midgen shared their views on live exports in an open letter to Prime Minister Jacinda Ardern.
"We feel strongly that a collective voice from the leading animal welfare and the leading animal rights organisations in New Zealand is the best way to make our voices heard. We want to see live exports banned effective immediately and we're proud to stand with SAFE to call for the law to be changed."
