The recipe calls for cooked wild rice, added at the beginning. I cook the wild rice in my Instant Pot during the day, using the timer delay, so it's done about the same time as the soup, then I add it.
You could add raw wild rice to the soup but you'd need to add quite a bit more broth. It absorbs about 3x its volume in liquid, usually, and takes over 60 minutes to cook simmering on the stove, about 35 minutes in a pressure cooker. I'm not sure how long it would take in a slow cooker. It's possible it might not even be done after 8 hours on low.