Scrobbling isn't officially supported by Pandora at the moment, however you can scrobble the music you listen to on the Pandora website by using a third party browser extension, such as the Web Scrobbler:
- Web Scrobbler for Chrome
- Web Scrobbler for Firefox
- Web Scrobbler for other browsers (Edge, Opera, Vivaldi etc.)
Also external Android scrobbling apps should be able to scrobble Pandora like Pano Scrobbler and the offiicial Last.fm Android app. Pano Scrobbler is also available in a Windows and Linux version now which might work with the Pandora desktop app:
The new iOS version of the Web Scrobbler for Safari should also scrobble the Pandora web player, but there is no solution for the Pandora iOS app or their desktop app. This would have to be implemented by the Pandora developers as an internal scrobbling option, there are several related threads on their support forum.
Also note that Pandora.com is a subsidiary of SiriusXM and only available in the US.