The card type choice is a replacement effect, not a triggered ability. It doesn't use the stack and can't be responded to. Notably, if you choose instant, your opponent can't try to cast an instant on the Emissary before it has protection.

You may choose any card type. The ones that may appear in a normal game of Magic are artifact, battle, creature, enchantment, instant, kindred, land, planeswalker, and sorcery. Supertypes such as snow and basic may not be chosen. Subtypes such as Angel, Aura, and Forest also may not be chosen.