They seem to work on this site okay, but on my installation I just get a link. It's a default installation and all other oEmbed providers seem to work okay, just not Vine. (Vines embedded in twitter links work okay for example, just not vines directly from vine.co)

Is there anything in the ACP I can check?

I'm on the latest version 4.1.15.

Someone should check out if the JavaScript call is made when you paste the URL and what the response from Vine is. Your server might be blacklisted or something like that. 

Thanks for the reply! Are there any developer tools I could use in Safari/Chrome to look at the Json response?

Ok I've checked it using the network tools in Google Chrome, if I post twitter/facebook/youtube etc I see lots of js responses, however if I paste a vine.co address I get nothing. It's like it's not even making the call.