Go to Prefs/Internet/custom web links. In a name box enter something like "Last.fm Artist"; in the web-link box, enter "https://www.last.fm/music/
<Artist>" (no quotes). Save.
Now go to Prefs/Hotkeys, and scroll down to the "Web: Open link (#)" with the number that corresponds to the order of your new custom web link in the last section. Assign a hotkey to it.
Now when a track is hilighted in MusicBee, hit the Last.FM hotkey you made, and your browser will open to the track's artist page in Last.fm.
Tested and working fine with Firefox 92.