The only (reliable) option to connect two or more audio apps in ios is via
Audiobus.
You can see compatible apps in
https://audiob.us/apps/. You might look for input applications. Sadly I can't see any podcast player with audiobus support but an app like
Amplifind can play from different sources (including spotify, itunes and mp3 files) and you can use another app, like Garageband, to record the stream.