If all of your data is public, you can use Vimeo's Simple Api.
The documentation for finding the videos in a channel is here : https://developer.vimeo.com/apis/simple#channel-request
Lets say you wanted to find the most recent video published to the VimeoHQ channel.
- Pull down the JSON from
http://vimeo.com/api/v2/channel/vimeohq/videos.json
- Parse the JSON into an array
- The newest video is the first element of the array
- Use the "url" field with oembed to build your embed link.
If your data is private you will have to use Vimeo's Advanced API.
This method is a little more complicated, and requires you to use OAuth 1.0. I recommend you use an existing library, and explore the vimeo.channels.getVideos method.