There isn't even agreement on when or where, little alone who. Probably some Neanderthal (literally).
Human speech is estimated to have started somewhere between 65,000-100,000 years ago. Theories differ. There's not even agreement on where to draw the line between words and sounds with meaning. There's no definitive way to prove anything it was so long ago. Writing, on the other hand, is relatively new, having started only (roughly around 8,000-10,000 years ago).
One consistent feature of any language, is that it evolves. New words are constantly being invented, old words drop out of usage. Those words are created by the speakers of a given language (or sometimes borrowed from another language).
So I wasn't joking when I said "humans" - assuming you don't want to argue the difference between Neanderthals and Homo Sapiens.