I'm not sure whether it was possible for the private companies contracted by the space program to patent their publicly-funded work.
Interesting to see what happens now that private enterprise is running its own programs alongside the government. I believe there will be a distinct chilling of relations between the main private players in a few years, once their main competition becomes each other rather than the government.
I am not sure it is possible for private companies to patent rocket work, with help from the public or otherwise. Rocket technology is heavily export controlled.