http://galileoandeinstein.physics.virginia.edu/lectures/srelwhat_files/image017.gif
(source: virginia.edu)
In both reference frames, there is an event where the light strikes the center of the top mirror.
In the frame in which the mirrors are at rest, the light has a vertical path.
Is it not clear that, if the mirrors are moving in a reference frame, the light path cannot be vertical?