Yes there are some records of the Palmer family in the Western Pacific Archive. There is a section listing all the applications for British passports or certificates of identity, from people who wished to travel, to the British Consular agent. They run from 1916 to 1964, and many have photographs of the person attached to the application form.
These records have been indexed, by me and a friend, and there are 4 entries for Palmer;
Alice Palmer in 1916, born on Waiheke Island.
Evelyn May Palmer in 1916, born in Ha'apai, Tonga.
2 for William Walter Palmer, in 1919 and 1922. He was born in East Tamaki, Auckland.
The 1916 applications do not have photographs, but WW Palmer has a photograph in 1919.
I am able to buy copies of these applications and the photo if you want. My email is firstname.lastname@example.org