Speedrun the event track
Completing an assignment in Vanguard progresses the "Complete 6 Vanguard expeditions" by one point.
There are 2 Vanguard Expeditions which have very easy/fast assignments.
Log in all 3 EVE characters on the account you want to progress the EVE event track for, BEFORE you do the assignments in EVE Vanguard. Open the event track with each character. After that, the EVE client can be closed. This ensures that all 3 characters can be progressed at the same time!
To begin, accept the "Avalon Rises" Expedition in EVE Vanguard and activate the first assignment which requires you to kill 3 NPCs and loot 6 boxes. Once the assignment is done, extract/die and complete assignment in the warbarge. Now ABANDON* the expedition and accept it again. Rinse repeat until you did it 6 times. Then log in to the EVE characters to claim the task. Do that 4 times in total and all 3 EVE characters will have completed the event.
There's a secondary Expedition that can help speed this up even more. Looting the boxes may award "Intel Fragment". Expedition "Beneath The Garden" has an assignment turning in these fragments.
Intel fragment visual:
