Donkeys were introduced to the Island of Hawaii by immigrants to the Islands who found them useful as beasts of burden, especially for farming and ranching work.
Donkeys were introduced to the Waikoloa area starting with 30 in 1975. There is lore about who moved donkeys, when and where, but it mostly conflicts and isn't going to turn up a likely party to help correct the situation.
There are now about 250 donkeys on the 20,000 acres between Waikoloa and Kamuela because they thrive in exactly the environment they have on this slope of Mauna Kea. They have adequate food and water, from nature, and their hooves need the lava rocks for grooming.