The horror and true-crime series “Last Podcast on the Left” played 39 live gigs last year, from Atlanta to Vancouver. Fans waited for hours to meet the three hosts, sometimes presenting them with “gifts” such as vials of blood and a relative’s cremains.

“It’s not ‘Prairie Home Companion,’” said the trio’s booking agent, Josh Lindgren, whose agency Billions represents bands like Arcade Fire. “It’s the only podcast show where I’ve seen a fight break out.”

Podcasters...