There is no prerequisite mission for accessing Vetra’s main loyalty mission, but you’ll need to have progressed through the story until you’re on the main mission Hunting the Archon.
Vetra will email you with news from her sister, Sid. Take the call with Vetra and Sid on the Tempest, which will begin the main loyalty mission Vetra Nyx: Means and Ends.
You’ll be tasked with travelling to a large meteorite in the Remav system in order to save some kidnapped colonists being held in an abandoned mine.
Head to the meteorite and drive the Nomad towards the waypoint, where the mission will begin once you’ve entered the mine itself.
The mission is a fairly straightforward combat sequence where you have to break out of a prison, followed by a boss fight at the end. Once you’ve done this and talked to Sid, Vetra’s loyalty will have been earned.
The process for Cora’s loyalty mission initially kicks off after you’ve established a settlement on Eos, at which point she’ll send you an email notifying that she has received word of a group of Asari refugees on Eos.
Go talk to her on the Tempest, and she’ll ask for your help in going to find them, which begins the mission Cora Harper: Asari Ark.
Head to Prodromos on Eos and talk to Hydraria, who will point you in the direction of a ship called the Periphona, which is located somewhere on the ice planet Voeld.
Take a trip to Voeld and drive far out into the eastern section of the map, at which point you’ll have to figure out which signal is coming from the Periphona (it’ll be the one situated in the north). Explore the wreckage, and this will put the mission on hold until you’ve completed A Trail of Hope.
At this point, Cora will send you another email, and if you talk to her on the Tempest it will initiate the main loyalty mission itself, Cora Harper: At Duty’s Edge which takes place on the Asari Ark. Complete this section, and Cora will be a fully loyal crew member.