Mystery Island, the tiny islet off Aneityum in Vanuatu's Tafea Province, has no lounge, and realistically it never will, because there is no terminal building here to hold one: AUY is a grass airstrip mainly used for the occasional charter, while most visitors arrive by cruise ship tender rather than by air at all. Anyone searching aneityum airport lounge or mystery island lounge is really asking about a beach landing point, not a departure hall.
The reality on the ground at Mystery Island is a cruise-day-stop experience: tender boats bring passengers ashore to a beach, a handful of local market stalls, and simple facilities built for a few hours of shore time, not an air terminal with seating, food service, or card-network access. There is no airside waiting area, no check-in counter with a lounge upstairs, nothing resembling the infrastructure a Priority Pass or airline club would need.
For genuine lounge access anywhere in Vanuatu, the relevant airport is Bauerfield International Airport in Port Vila, the country's main gateway on Efate. Port Vila's terminal is modest by international standards, without the large multi-brand lounge networks found at bigger Pacific or Asian hubs, but it is the only place in the country where an actual airline or premium-cabin space might exist, and where any real access question, airline status, a business ticket, should be directed.
If your visit to Mystery Island is part of a cruise itinerary, plan around the ship's own facilities for comfort during the call rather than expecting anything shore-side, since AUY was never built as a passenger air gateway. If you are instead flying within Vanuatu and passing through Port Vila, confirm current lounge or premium-service arrangements directly with your airline before you travel, since a small national carrier's facilities can change without much notice. Either way, Mystery Island itself remains a shore excursion, not an air-travel waiting room.