I read an article recently by Common Sense Media and they had a list of the top 20 recommended podcasts for kids (some of which are created by kids!) My favorite for my 7-year-old is the bi-weekly Book Club for kids.
Thank you both for the suggestions. Charlie, I meant "like" as in "similar to". Basically, a fictional story that continues every week rather than a one off. I should have explained my request much more clearly, but the link you posted lead me to tons of great podcasts so thank you for that.
Pique, I found the article you mentioned and you're right, it's a great resource. Thanks for sharing it!