Last month, Magnum reported the death of member Martine Franck, widow of Henri Cartier-Bresson, a founder and its most famous member. The organization now has no direct link to its patriarchs for the first time in 65 years—and operates in a world of journalism far different from the one Magnum confronted in 1947.

In that year, four photojournalists got together to report on stories that the conventional press might not care about, and to share the profits. They called themselves Magnum. Over the years, more photographers were...