Viewing Vine videos in third-party Twitter clients isn't the best experience right now: they work, but you have to watch using an external player after following a link, whereas the official app displays them inline. From what we're seeing, though, it looks like it may not be too hard for third-party developers to integrate the new functionality in full.
Paul Haddad of Tapbots, creator of best-in-class iOS client Tweetbot, has tweeted a screenshot that shows Vine videos appearing in the main Twitter feed of his app. This suggests that, unlike other new Twitter functionality such as the Discover feed, it's not being blocked for use by third-party clients. Of course, this is just a proof of concept for now, but hopefully we'll start to see Tweetbot — and other apps — adopting Vine in full soon.