Team citations are earned in two ways during co-op mode. Some are mission-specific, for example, completing a mission in below a time limit, others are awarded when the entire team earns the same individual citation.
At this point in the game's life, you can usually find a random group racing through missions during the weekends and afternoons. Such groups will almost always earn time awards. Completing missions quickly is the most efficient way to level characters, so you'll probably get many then.
Individual citations are
Medic - Performing 2 revives
Headhunter - Headshots
Vulture - Scavenging parts/ looting corpses 10 for Bronze
Ruthless - Kills
Support - Assisted Kills
Hacker - Hack at least 3 enemy devices eg wolverines, armoured car
No parking - destroy at least 3 armoured vehicles
TakeDown - At least 3 Takedown
Brew Up - Construct at least 3 GTK items during mission
Drone Control - Hack 3 Drones
Weapon Citations - 10 Kills with 1 weapon
Of these, vulture, ruthless, and support can easily be earned by all members of the team. Pay attention to what your teamates have and try to earn them yourself.