NASA didn't build the rocket. Private industry did. You're confusing socialism with capitalism again. In any case would you want to fly to the moon in a rocketship built by the lowest bidder?
Some interesting facts about who “built” the rockets and such that went to the moon on Apollo 11.
The spacesuits were built by Playtex of bra fame.
The Lunar Module was developed by Grumman Four subcontractors were involved. Bell Aerosystems, Hamllton Standard, Marquart and TRW Space Tech.
Primary guidance system was developed by MIT. The guidance computer was built by Raytheon. The Abort guidance system was developed by TRW.
The Command Module was built by North American Aviation.
Stage 3 of the Saturn Rocket was built by Douglas Aircraft Company
Stage 2 was built by North American Aviation.
Stage 1 was built by Boeing.
Now I guess one could try and argue that NASA stacking all of these completed components together in the VAB means NASA 'built" the spacecraft. Thats quite a reach.