This was once possible when BandCamp still had RSS feeds.
They disappeared ca. 2 years ago.
It seems the only workaround for now is following artists.
Just found a thread on Discogs with more discoveries about dates hidden in page source.
• publish_date is null (and the draft only visible to you) until you click the Publish button, then it is when you clicked that button; it never changes.
• new_date is when you first clicked the Save Draft button, and it does not change.
• mod_date is the last time you clicked the Save Draft or Update button; it’s when you last made any change.
• release_date is the ISO 8601-formatted version of the mm/dd/yyyy value you put in the release date field.