The jump ship is presumably built along with the Waverider. As the recruitment of the Legends in 2016, the Legends frequently used it to travel across time and space separately or without being noticed.
In 2018, John Constantine, along with Charlie and Zari Tomaz who disapproved this action used it to alter the timeline for own purpose as they finally gave up due to the timeline gets worse every time they alter.
The original jump ship has 8 seats along with a pilot suit in the front. This variation had a door opened on the right side on the back and 2 pair of landing gears attached to the front and back of it.
At an unknown time, the Legends either made a new jump ship or modified to former one. The new variation had the same amount of seats. The difference are the way how the door opens and the landing gear has the same design as a helicopter’s landing gear.
In 2018, the Legends got a different jump ship which had 2 doors on the sides and 1 on the back with the same landing gear design as the previous one. However, it is unknown if this was the same previous one.
DC's Legends of Tomorrow