In 1899, what would the logistics be for a group of Americans to travel to Tahiti and buy land, and emigrate there?

The easy part is getting there! In 1899 the only predictable link between Tahiti and the US is the Union line running San Francisco-Papeete-Auckland. So as long as you could get yourself to San Francisco getting passage to Tahiti would be pretty straight forward-- alternatively you could possibly find a cargo ship under contract to carry lumber or something from San Francisco to Tahiti and get a berth on it. Or if it is late summer, its possible there might be a few captains preparing to head down to the Tuamotu archipelago (right next to the Society Islands/Tahiti) for the pearlshell fishing season and who could be persuaded to take you down. If you are really wealthy, you could charter a smaller ship and crew for yourself and take it over. Otherwise, the Union line is the most predictable and obvious choice.

Buying land is going to be more difficult, in 1899 most land has never been officially registered under the French Civil Code and remained in the hands of Tahitians. The French had, since 1866 during the years of the French Protectorate, been attempting, somewhat halfheartedly to institute a European style private property regime. Attempts had largely failed with the exception of land in the port city of Papeete and then a few large plantations like Atimaono or Opunohu over on Mo'orea. Tahitian land tenure was nothing like that of France in the 19th century; land was not owned by individuals, but rather individuals, families, or clans maintained various rights of access or denial over resources. For example, and one that I am simplifying quite a bit, a high ranking ari'i might always have access to a particular grove of 'uru (breadfruit) because of his rank and position, but this would not be the case for everyone else in his family or his clan. He would have the right to restrict access through a rahui, a prohibition on a place, resource, activity for a set amount of time. Likely he would not be the only one with this power and there may be trees within the grove which were exceptions to his authority. Indeed, while he might govern access to the grove, he might not have the right to take 'uru from it, that could be the sole right of a different family, or clan, or individuals of a certain rank. So for the French, the question was who owns the land? They tended to view this through the lens of French feudalism and translate the ari’i as lord who could dispose of property as he saw fit. Yet, republican French sentiments also recognized that this was far from the case on the ground and sometimes construed land tenure as communal in nature, which meant the community/family had to agree to invest ownership in individuals (or establish fractional ownership which was either seen as not ideal because the land was more difficult to sell; or seen as ideal because it meant the land was more difficult to sell and therefor could not be entirely alienated to Europeans— two sides of the same coin essentially). What often happened is that attempts to transform indigenous land rights into French private property involved lengthy land court cases. The only way to for sure buy the land was to buy out anyone who had a claim to the land so they would not object in court.

The French Government recommended to emigrants that they not try to buy land as the plots were too small and instead lease land. Land leases held up better in courts as the question of rents could be argued between all those who held a valid claim on the land, while you, the renter could go about your business. In 1899, you arriving in the midst of a big push by the government to register land and conduct land surveys, ultimately they won't make a whole lot of progress outside of Papeete, but at least in town there should be plots of land available.

As to staying as a permanent resident. You would definitely need to meet with the French governor, present your funds/resources (justify being allowed to reside and conduct business in the colony). You might want to make friends with some high powered lawyers like Auguste Goupil who can help you get access to land and laborers if you need. If you want to become French, that is also an option, though I am not entirely sure of citizenship requirements in 1899. I believe there is an American consul in 1899, Julius Dreher, who could also help you with questions of residency or contact back to people within the United States.

*edit : cheers for the gold! As an historian it’s always amazing to get any sort recognition.