So, the Channel Images input for Wygwam works fine. We can add an image to the wygwam field, save it, and voila. When we edit the entry though, click the wygwam to edit, the images disappear. Not just visually, but also in code.
All modules are up to date (Wygwam is 3.3.3 and Channel Images is 5.4.19). Expression engine is 2.9.3