Wikipedia says that the song first appears in 9th Century Seder Rav Amram. I don't have access to that Sefer, and I couldn't locate anything about who wrote this. Was it Rav Amram, himself, or someone else?
Louis Finkelstein ZTL, one of the greats in the field of academic study of Judaism, analyzed dayeinu, and to the best of my knowledge his analysis has gone unchallenged in the academic world. (Pre-Maccabean Documents in the Passover Haggadah, Harvard Theological review, 35:4 and 36:1). In particular the Goldschmidt haggadah and most modern academic commentaries I've seen cite Finkelstein without challenging his claims.
Finkelstein dated Dayenu to the second temple period prior to the Hasmonean revolt. His analysis was based on the fact that the song does not mention the Hasmonean revolt, the destruction of the temple, or the coming of mashiach, all of which would be mentioned had the song been written at a later time. The fact that the song ends with the building of the temple implies that the author was writing during a time when the temple still stood.
As for the author, I suspect the age of the piyut makes it impossible to ever say with certainty. However, Finkelstein suggests that the focus on the temple indicates the author was likely a priest who saw himself, a descendant of Aaron, as a rival to Moshe, who is mysteriously absent. He even goes so far as to suggest the author was the high priest Jason, although you will have to read his whole paper to see his logic.
As wikipedia says, it didn't enter the haggadah until much later. The first time it appears in print is Rav Saadia's siddur where it is printed as an appendix and is optional (Goldschmidt haggadah).
According to the Hebrew Wikipedia article, the author of "Dayenu" is unknown, and, as you stated, its first appearance is in the Siddur of Rav Amram Gaon in the 9th Century.
לא ידוע מי חיבר את מילות הפיוט. הפיוט אינו מופיע במשנה, בתוספתא, בתלמודים ובמדרשים, ואף לא בנוסח ההגדה של הרמב"ם. מאידך, הנוסח מופיע בנוסחים קדומים של ההגדה, כגון ההגדה המובאת בסדר רב עמרם גאון, וכן בספר הכוזרי ג, יא
The author of the piyut is unknown. The piyut does not appear in the Mishnah, Tosefta, Talmud or Midrash, and not even in the Haggadah of the Rambam. On the other hand, the text appears in early versions of the Haggadah, such as the Haggadah presented in the Siddur of Rav Amram Gaon, and also in Sefer HaKuzari (3:11).
As mentioned by @NoachMiFrankfurt in the comments, many piyutim are of unknown authorship, so this does not come as a surprise.