Prequisite: This becomes available after “Nexus Reunion”, after Tann tells you about your new assignment. You can ask him more about the missing arks.
This is a long-running task which the objectives will be unlocked as you play the game. They're also somewhat tied to other tasks and loyalty missions so doing those specific optional missions will help complete the individual tasks. As such, this is something that you don't need to actively do something to complete.

Refer to the details on Cora's loyalty mission "Cora Harper: Asari Ark". After finding the wreckage of the survey ship from the ark in Voeld. Continue with the story missions until you receive an email from her stating that she was able to locate the ark. This starts her next loyalty mission, "Cora Harper: At Duty’s Edge". The mission will take you to the ark itself, clear its Kett occupants, and restore the ark back to its normal functions.

Refer to the details of the Allies and Relationships mission "Turian Ark: Not Dead Yet". After meeting up with the turians in Havarl and speaking with their leader, Avitus Rex, continue with the story some more until you have access to Eladaan. Avitus Rix will send you an email about the reported turian ark pieces spotted in the planet. This completes the previous AR mission and starts the next one "Turian Ark: Lost But Not Forgotten". Reading this email will complete the objective for this mission. Finding the actual ark is part of the aforementioned AR mission.

Unlike the other two arks, the Salarian ark can be found and boarded during the main story mission “Hunting the Archon”.