In case it helps anyone, Vimeo have added a ‘background’ parameter for embedding videos, that autoplays videos silently. In some cases that will be useful where people want to mute videos – this is their example:
<iframe src="https://player.vimeo.com/video/76979871?background=1"
width="500" height="281" frameborder="0" webkitallowfullscreen
mozallowfullscreen allowfullscreen></iframe>