I have occasionally heard "blame me/them on it" and thought it was a bit jarring. The more common ways, I think, are "blame me/them for it" and "blame it on me/them."
Alex said in a text message. “He’s probably going to blame me on it, but deep down, he knows he did it.” (source)
I told you he was going to blame me on it. (American TV series House of Cards)
My question is about the register and correctness of this phrase.