NASA's SoundCloud channel has something for you. On NASA's SoundCloud channel you will find audio from Apollo, Mercury, and Discovery missions. You'll also find audio of rocket sounds and space sounds. The set of recordings of most interest to me is the set of three audio recordings of President Kennedy which includes his famous "We Choose the Moon" speech.
Applications for Education
NASA's SoundCloud channel could be a good place to find audio to support lessons on the development and evolution of the space program. The recordings are in the public domain. You and your students can download the recordings to re-use in video productions about space. I might have students use the Kennedy recordings in a video project about American culture and the Cold War in the 1960's.
If you cannot access SoundCloud in your school, most of the recordings are also available on the NASA Sounds website.
H/T to Open Culture.