u/snapshot52 might be able to answer more specifically about how genuine Native American stories and beliefs have inspired or been appropriated by European invaders, I'd like to touch on the question of more recent appropriation - that is, the use of Native Americans as a potential source of lore for horror stories, regardless of whether or not there any actual Native American folklore involved.

The Americas were regarded by Europeans as the "New World" - not because they did not have any history of their own, but because they had no history that the Europeans chose to recognize; by force of arms they made and broke numerous agreements, pushing Native Americans back from their lands, massacring them either with brutal violence and open warfare or slowly through starvation, restriction, and cultural assimilation.

Yet the Native Americans were recognized as have preceded the Europeans on the continent, and so in the absence of castles, ruins, and ancient history that served as the inspiration for many horrors in European folklore, by default any "old" horrors in the Americas pretty much had to come from Native Americans, at least on a hypothetical level - and so Native American horrors became a literary device which many authors, particularly in the pulp period, could use as a way to frame a horror story. Even if said story contained no actual Native American folklore, the claim lent verisimilitude to the tale.

u/sunagainstgold and I talked about this a while back in When and how did "Indian burial ground"? become a horror trope? Were Native American burial sites and rituals in any way scary and mysterious to the colonists or is this a cliché born in 70s/80s, with Stephen King helping to spread it?, but you can see it in stories like Robert E. Howard's "Old Garfield's Heart" (Weird Tales December 1933) where he writes:

"I knowed you wouldn't understand," said old Jim. "I don't understand myself, and I ain't got the words to explain them things I feel and know without understandin'. The Lipans were kin to the Apaches, and the Apaches learnt curious things from the Pueblos. Ghost Man was—that's all I can say—alive or dead, I don't know, but he was. What's more, he is."

In part, this is a deliberate reference to the characterization of Native Americans into specific roles in American fiction - as helpers or opposition to white people, sometimes spiritually attuned or mystical. African-Americans and Asians faced similar "exotification" and stereotyping, most often in a derogatory fashion, but it was the common default perspective in American literature to assume that white people were scientific or rational, and comparatively lacking in the "superstitious" cultural upbringing that went with indigenous horrors - and so were often relatively powerless against them, unless instructed and helped by someone who did know about them.

So if you look at a story like Algernon Blackwood's classic "The Wendigo" (1910), the eponymous horror has very little to do with genuine Native American folklore - but Blackwood doesn't care! He's using the idea of this being a creature of Native American folklore in order to increase the audience's suspension of disbelief.

"And what's the Wendigo, pray?" Simpson asked quickly, irritated because again he could not prevent that sudden shiver of the nerves. He knew that he was close upon the man's terror and the cause of it. Yet a rushing passionate curiosity overcame his better judgment, and his fear.

Défago turned swiftly and looked at him as though he were suddenly about to shriek. His eyes shone, but his mouth was wide open. Yet all he said, or whispered rather, for his voice sank very low, was: "It's nuthin'—nuthin' but what those lousy fellers believe when they've bin hittin' the bottle too long—a sort of great animal that lives up yonder," he jerked his head northwards, "quick as lightning in its tracks, an' bigger'n anything else in the Bush, an' ain't supposed to be very good to look at—that's all!"

"A backwoods superstition—" began Simpson, moving hastily toward the tent in order to shake off the hand of the guide that clutched his arm. "Come, come, hurry up for God's sake, and get the lantern going! It's time we were in bed and asleep if we're going to be up with the sun tomorrow...."

This kind of thing can send people off into wrong directions, looking for Native American sources for purely fictional literary inventions. For example, there are several articles trying to track down the figure of "Yig," supposedly a Native American serpent-deity of the Great Plains who was identified with the Aztec Quetzacoatl - but Lovecraft made up "Yig" out of whole cloth in "The Curse of Yig" (Weird Tales Nov 1929), which he ghostwrote for Zealia Bishop. Another such character is the shaman Misquamacus, an "ancient wonder-worker" of the Wampanoag - who featured in August Derleth's novel The Lurker at the Threshold (1945), and was then used by Graham Masterton in his novel The Manitou (1976), which inspired both a 1978 film and a handful of sequel novels - but both cases were drawing off an unpublished fragment of text of Lovecraft's written in the style of an entry of 17th/18th century colonists like Cotton Mather entitled "Of Evill Sorceries Done in New-England, of Daemons in No Humane Shape."

The fact that there was no Misquamacus didn't stop anyone at any point, it was the idea that there was some genuine Native American folklore at the base of it that was sufficient for everyone involved to pick up the ball and run with it. So in that sense, the appropriation has been relatively ongoing from a very early period, because people wanted ancient, local horrors - but they either couldn't find anything suitable or didn't want to be limited by genuine beliefs.