As others have noted, NASA's rockets have largely been designed, built, and operated by private contractors for its entire history. The difference was that those projects were managed by NASA via cost-plus contracts, often at mind-boggling cost.
The crowning achievement of this system was the Apollo program. Which was followed up by the Space Shuttle, a system so expensive and full of compromises that it wasn't really good at much of anything except freezing the state of US space travel at the technological level of 1976 for thirty years.
NASA doesn't design and build its own one-off plane when it needs to send some people to a conference in NYC; they buy them commercial airline tickets. Likewise, purchasing launch services from a commercial provider is far cheaper and more effective than engaging contractors to build a rocket to spec.