Does your CS team record their calls? If so, you can ask them to give you a heads up when a customer tells them something that worked really well. You can then take the transcript of the calls and build it into a customer story, then reach out to the customer and ask if they'd mind if you published it. That's a super low lift for them.
Depending on your customers, you can also package telling their story in a way that puts them in a good light. For example, you could boost the LinkedIn posts that they make about the customer story, which gives them a lot of brand exposure for no cost and little effort. And you make sure the stories are about them doing something awesome (with you) instead of all about you.