tl;dr - none, you wouldn't even know who the Tu'i Tonga was.

I love the question, there is something romantic imagining vast oceanic empires. However, there are a few assumptions of familiarity here that just don't quite translate. The term fisherman assumes a sort of professional status, or a professional identity, that just did not exist amongst people in 15th century Vanuatu-- there are distinctions made in some contexts between coastal people and interior ones-- which might be the distinction you are in fact drawing up. While coastal people did travel between islands, people in the interior generally would not have done so. Most of the islands in Vanuatu (which is not an island itself, nor geographical feature, its is at best borders inherited from the Condominium of France-Britain that encompass some 80 islands), do not have this distinction between coastal/interior population due to the size of the islands.

The Tu'i Tonga did not really exercise control or authority outside of Tonga, Samoa, and the Eastern part of Fiji--so no where near any of the islands of Vanuatu. Even within this region, the Tu'i Tonga would not have exercised control over the daily lives of most people, rather their local chiefs and priests would have been the ones with which they interacted. The Tu'i Tonga was a prestigious title, but the actual exercise of authority varies significantly according to oral traditions and eventually declines in prestige before lapsing completly. The term Empire is problematic, because it wasn't a singular political entity that we might recognize, the Tu'i Tonga was at/near the pinnacle of a prestige network but did not rule with absolute power. Nor was the goal of the Tu'i Tonga to extract goods from outlying archipelagos for their monetary value, there was no sort of abstract market value attached to goods. Malama Meleisea characterizes the 'empire' more as the center of a prestige network held together by genealogical links and a strong Tongan voyaging tradition.

For many centuries Tonga, where natural resources were few, acted as a 'middleman' between Samoa and Fiji, the sources of red feathers, timber for boat building, adze blades and fine pandanus cloth.