Use of blame in sense 2b with on has occasionally been disparaged as wrong. Such disparagement is without basis; blame on occurs as frequently in carefully edited prose as blame for. Both are standard.
Examples of BLAME
Don't blame me. You are responsible for your own problems.
My father always blames everything on me.
I blame the poor harvest on the weather.
Origin of BLAME
Middle English, from Anglo-French blamer, blasmer, from Late Latin blasphemare to blaspheme, from Greek blasphēmein