You might want to try asking this over at /r/spaceflight or /r/space as well.

I'm not a historian, but I am a space engineer, and familiar with the history of the space race. The question of who is most responsible for various contributions is a rather complicated and at hotly debated topic. One of the more notable, and even controversial figures is Werner von Braun, the most well-known beneficiary of Operation Paperclip.

One major thing to point out, though, is that the space race happened in the midst of numerous other historical developments and happenings, and not in a vacuum. In particular, in the 1950s several important scientists and engineers were forced out due to the anti-Communist fervor now commonly referred to as "McCarthyism". This includes people like Frank Malina, one of the founders and early directors of the Jet Propulsion Laboratory (now NASA JPL) and Que Xuesen, who was also "present at the creation" of JPL, among others. As such, whatever the contributions of former Nazis to the space program in the US, it's important to remember that these people were both filling in gaps being created by anti-Communist purging, and also given greater opportunity and prominence due to the same.

Von Braun's contribution to NASA and the space program is much touted in official NASA history, but it's impossible to say what would have happened if he wasn't given the room to advance that he was due to vacancies above.

EDIT: I recommend "Astro Turf: The Private Life of Rocket Science" by M.G. Lord for discussion of McCarthyism's effect on the space program in general and Malina's exile from NASA, in particular.