Apple has made it possible to listen to Apple Podcasts on the web. The Movies and TV show web interfaces have also been updated (via Mac Generation).
Apple Podcasts Web Player
Apple is slowly killing off iTunes, with the company bringing iOS TV, Music, and Podcasts to macOS via Marzipan. That death includes overhauling each app’s web interface.
Apple Podcasts on the web offers a similar experience to the app. Episode titles are featured in bold, with a description below. Each episode also has an individual page, where you can see full show notes and other details.
Although there is a prominent button telling you to listen in iTunes, a noteworthy aspect of the new design is that URLs now use podcasts.apple.com, instead of the old itunes.apple.com.
See for yourself how it looks on the Mac.
Now all we need is an official Apple Music web player (there are plenty of unofficial ones).
Further Reading:
[The Breakup of iTunes Looks to Be Upon Us Soon]