How do you get proof of the cryptids on Cryptids Island?

Answer 1

You never do. You will, however, find substantial proof of the similar creature, Bigfoot.

At the monastery in the Himalayas, the monks are unwilling to part with their Yeti scalp. However, you do get a camping lantern, and one of the monks lets you climb up for a photo of a mysterious footprint in the snow. (You will need the camera from Loch Ness.)

When you have proven the other three cryptids (Nessie, Chupacabra, and the Jersey Devil), Mews will send you to the Pacific Northwest in pursuit of Bigfoot.

Answer 2

You need an actual photo of Nessie, a tooth from the Chupacabra, and egg shells from the Jersey Devil. (You do not prove the Yeti, because all you can get is a photo of a footprint.)

However, when you have proven those three, Mews will send you to hunt for Bigfoot instead.

(see the related question)