Do onomatopoeias have etymologies?

It’s a very insightful question, OP. If an onomatopoeia is a completely transparent mapping of natural sound to human language, then it is an inevitability, and there’s no point attributing it to one coiner or another, one language or another: the onomatopoeia is just there, a sound ready for humans to imitate, and humans will keep imitating it over and over in the same way.

… Except, well, no. The mapping of natural sound to human language is not inevitable: if the sound isn’t articulated by a human mouth as communicative language, then there’s no single deterministic way of representing it in language. Different languages can come up with different ways of mapping it. See the renditions in diverse languages of cock-a-doodle-doo – Wiktionary.

(And I strongly suspect Middle English had a different version of “cock a doodle doo”, Andrew Dunbar, but I can’t find it: Chaucer’s Nun’s Priest’s Tale has just “Cok! cok!”)

And onomatopoeias can be borrowed between languages too. The Turkish onomatopoeia for sneezing is hapşuu. The Greek onomatopoeia is apsu. Which is exactly what you’d get if you borrowed the Turkish onomatopoeia into Greek as a normal word, and accommodated it to Greek phonology.