Yes, there is a service that automatically puts your live stream (from living as one or any other live stream provider) and Vimeo into one json feed for Roku and updates the feed every 24 hours.
It's called Tappla and costs $ 24 / month. It syncs with Vimeo via their API every 12-24 hours to add new videos from Vimeo to your feed. You can define what Vimeo videos, showcases, playlists should be synced and added to the feed.
For the live stream, you simply copy and paste the HLS streaming URL into your Tappla dashboard.
If you have more questions or need help, just send me a message (I'm one of the founders of Tappla).