Which first all-private astronaut mission visited the ISS in 2022?

Axiom Mission 1 was the first all-private astronaut mission to visit the ISS in 2022. Also called Ax-1, it launched on April 8, 2022, aboard a SpaceX Falcon 9 carrying the Crew Dragon Endeavour. Axiom Space organized and operated the mission, while SpaceX provided the spacecraft and launch system.

The four-person crew included former NASA astronaut Michael López-Alegría as commander and private participants Larry Connor, Mark Pathy, and Eytan Stibbe. They docked with the station on April 9 and conducted research, outreach, and educational activities during their stay. Bad weather delayed their return, extending the mission to roughly 17 days in orbit instead of the originally planned shorter visit.

Axiom Mission 2 was a later private ISS flight, not the first. Inspiration4 was also a private Crew Dragon mission, but it flew a free-flying orbital mission rather than visiting the ISS. Polaris Dawn likewise did not visit the station. Ax-1 therefore holds the specific record described in the question: the first fully private flight to the ISS.