Interesting puzzle that is quite hard to stop.
4 moves, as far as I can tell.
[FEN ""] 1. e3 e5 2. Ke2 Qh4 3. Kf3 d6 4. Qe2 e4#
[FEN ""] 1. d3 d5 2. Kd2 e5 3. Kc3 Be6 4. Qd2 d4#
And one more:
[FEN ""] 1. e3 e5 2. Ke2 d5 3. Kd3 Qf6 4. Qe2 e4#
Same theme for all of them, really.
I think it is 5 moves.
[FEN ""] 1. a3 e5 2. a4 e4 3. a5 e3 4. a6 Bc5 5. axb7 exf2#
You can't do it any shorter as you need at least 4 moves to get your pawn to the 2nd line where it can checkmate the white's king and one more move to bring a supporting piece (queen or bishop), so the white king wouldn't be able to capture your pawn.