Apple has announced a new feature adding transcripts to podcasts:
Apple automatically generates transcripts after a new episode is published. Your episode will be available for listening right away, and the transcript will be available shortly afterwards. There will be a short delay while we process your transcript. If portions of your episode change with dynamically inserted audio, Apple Podcasts will not display the segments of the audio that have changed since the original transcription. Music lyrics are also not displayed in the transcripts.