James T. Kirk
Officer shards can be earned by completing the following missions:

Combat Training 33, 36, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50: Veteran, Hold The Line, Interdimensional Fugitive, Specialist, Mirror, Mirror Part 7, and The Enterprise Incident. Also, shards can be gained through the Premium Officer Chest, the Ultra Officer Chest, the Federation Officer Chest, Crucible of War Officer Chest, and the Field Training Milestone rewards. Occasionally, more shards can be acquired through Rotational Ultra Chests. You can also exchange Transporter Patterns for extra shards. Extra shards can be acquired through the Offers tab.

Spock
Officer shards can be earned by completing the following missions:

 Combat Training 33, 36, 39: Rookie, Jellyfish Smack, Mirror, Mirror Part 7, Ripples in a Pond, and The Klingon Separatists. Also, shards can be gained through the Premium Officer Chest, the Ultra Officer Chest, the Federation Officer Chest, the Discovery Officer Chest, and the Crucible of War Officer Chest. Occasionally, more shards can be acquired through Rotational Ultra Chests. Extra shards can be acquired through the Offers tab.

Leonard McCoy
Officer shards can be earned by completing the following missions:

Combat Training 31, 34, 37: Rookie, and Mirror, Mirror Part 7. Also, shards can be gained through the Premium Officer Chest, the Ultra Officer Chest, the Federation Officer Chest, and the Crucible of War Officer Chest. Occasionally, more shards can be acquired through Rotational Ultra Chests.

Nyota Uhura
Officer shards can be earned by completing the following missions:

Combat Training 32, 35, 38: Rookie, and Mirror, Mirror Part 7. Also, shards can be gained through the Premium Officer Chest, the Ultra Officer Chest, the Federation Officer Chest, and the Crucible of War Officer Chest. Occasionally, more shards can be acquired through Rotational Ultra Chests.

Montgomery Scott
Officer shards can be obtained from the Premium Officer Chest, the Standard Officer Chest, the Federation Officer Chest, and the Crucible of War Officer Chest.

Hikaru Sulu
Officer shards can be earned by completing the following missions:

Mirror, Mirror Part 7, and Salvage Operation. Also, shards can be gained through the Premium Officer Chest, the Standard Officer Chest, the Federation Officer Chest, and the Crucible of War Officer Chest.

Arkady Ivanov
Officer shards can be earned through the Premium Officer Chest, the Standard Officer Chest, the Federation Officer Chest, and the Crucible of War Officer Chest.