(I'm using iOS8.1 iPad2)
(Not actually sure about Android, because e621.net has an unofficial app for that mobile OS.)

So when I load page with a HTML5/.webm video on e621.net the play button shows, but when I hit the/that play button nothing happens, even when I try to play it directly from the "Download" link.

Like I said in the title above, this only happens on e621.net and that all other sites that I have used/been on that have the support of HTML5/.webm for mobile, successfully works every time.

I thought the website..this website deliberately only allowed webm videos so it was instantly supporting mobile users.
(^That didn't meant to sound harshly at all^. ;) )

Bug reports go to https://e621.net/forum/show/28544?page=86.