To qualify for this timeline, the orbital rocket had to be designed, engineered and funded by the commercial organization. The rocket must also have successfully delivered a payload to orbit.
1990
April 5
🇺🇸 Orbital Sciences Pegasus
Orbital Sciences became the first company to launch a privately developed rocket into orbit, the Pegasus, on April 5, 1990.
1994
March 3
🇺🇸 Orbital Sciences Taurus (Minotaur-C)
Taurus launched on March 3, 1994, from Vandenberg Air Force Base and successfully reached orbit.
2008
September 28
🇺🇸 SpaceX Falcon 1
SpaceX’s first successful launch was performed on September 28, 2008, by Falcon 1 from Omelek Island, Marshall Islands.
2010
June 4
🇺🇸 SpaceX Falcon 9
Falcon 9 Flight 1 was on June 4, 2010, from Cape Canaveral.
2013
April 21
🇺🇸 Orbital Sciences Antares
First flight was on April 21, 2013 from Wallops.
2018
January 21
🇺🇸 Rocket Lab
Rocket Lab successfully launched its Electron rocket from Mahia Launch Center in New Zealand on January 21, 2018, carrying three cubesats into low earth orbit. This was the first time that a rocket entered orbit after launching from a privately owned and operated spaceport.
2018
February 6
🇺🇸 SpaceX Falcon Heavy
Falcon Heavy’s successful test flight took place on February 6, 2018, from Cape Canaveral; Elon Musk’s personal Tesla roadster was put into a solar orbit.
2019
July 25
🇨🇳 i-Space Hyperbola 1
Chinese private company i-Space successfully launched its Hyperbola 1 rocket from Jiuquan Satellite Launch Center and sent several small payloads into earth orbit on July 25, 2019.
2020
November 7
🇨🇳 Galactic Energy Ceres-1
China’s Galactic Energy successfully launched its Ceres-1 rocket from Jiuquan Satellite Launch Center into orbit on November 7, 2020, becoming the second Chinese private company capable of launching satellites into orbit.
2021
January 17
🇺🇸 Virgin Orbit
Virgin Orbit successfully achieved orbit on January 17, 2021, using their LauncherOne vehicle to deploy a payload for NASA.
2021
November 20
Astra Space Rocket 3.3
Astra Rocket 3.3 vehicle successfully reached orbit on November 20, 2021, after launching from Pacific Spaceport Complex Alaska carrying a payload for the United States Space Force.
2022
October 1
Firefly Alpha
Alpha successfully reached orbit on October 1, 2022, after launching from Vandenburg Space Force Base.
Sources: Wikipedia and company websites