The claim of the article is not about Oppenheimer's ideology. Rather, its claim is that he was at one point a member of a specific organization, the Communist Party USA, despite denying this later. His security investigation was something of a conflict between guys like Boris Pash, who saw communists everywhere and wanted to purge them, and John Lansdale, who defined a communist as "someone more loyal to Russia" than to the United States. This is described endlessly in "
Lawrence and Oppenheimer" (written in the 1960s) and the more recent Oppenheimer biography "Inside the Center" by Ray Monk. I expect American Prometheus goes into all of it too, but I haven't read it.
The Monk biography leaves open the possibility that Oppenheimer was once a CPUSA member and covered it up. He might have been, but his own hiring policy at Los Alamos was he willing to hire ex-CPUSA members as long as they were not current members.