I use my roku stick extensively to consume music through Pandora, youtube, somafm, vevo, and similar. I would love for these listens to be recorded to last.fm. I believe it should be fairly simple to integrate and as I understand, the last.fm api is free: https://www.last.fm/api
If you want a Last.fm app for the Roku, you should contact Last.fm and request they develop the app. Roku doesn’t write the apps for the Roku devices. They have a community forum of their own.
If you are able to use Plex for your music listening, you can connect your Plex account to last.fm. This enables scrobbling of any music you listen to on the Roku Plex app.