Of course, I'd still recommend that you use one of the many UWP podcatchers for Windows 10 Mobile instead. Even the new beta Podcast Lounge 2. But yes, if you don't mind coughing up a little to Shifty Jelly then you can use their servers to run a Pocket Casts 'instance' in a tab in your Edge browser:
Note that these solutions necessarily involve streaming everything, i.e. there's no download facility, though you can switch away from Edge (or even turn the display off) while listening, thankfully, so it's not as if the screen player has to be shown during playback. I did note the odd bug though and some podcasts had to be kicked off a couple of times before the audio actually started playing.
Talking of bugs, Shifty Jelly are still in my bad books for launching on Windows Phone 8.1 in a blaze of:
Welcome to the first update of many for our Windows app! Those of you who started countdown timers to see how long until we abandoned it are out of luck I'm afraid. It's nothing personal, but I'd avoid buying lottery tickets if I were you ;
... and then shipping one minor bug fix in three years. Very disappointing. I reviewed it here, back in 2015. Oh well.
At least the options are plentiful anyway. Pocket Casts in the browser, Pocket Casts in their old 8.1 application. Or use a modern UWP podcatcher. It really is up to you!