The first 4 boxes each week that you get from world activities will contain a key. World activities are twice-weekly zone assignments (the elite WQs unlocked by doing a few WQs in that zone), the Awakening the Machine weekly quest, the Kobold treasure digging weekly quest, Spreading the Light, the Azj-Kahet pact weekly quest, the Dornogal theater weekly quest, and the weekly in Dorn given by either Brinthe or Faerin. You do not need to do all of these each week, just any combination of 4 of those that you want to do each week.
You can get additional keys each week by getting 100 coffer key shards per week, which is easiest by getting a group to do a big Radiant Echoes pull, which are special events that cycle in the zones. You get Radiant Echoes by unlocking the chest at the end of a Bountiful Delve. Key shards can sometimes be dropped by certain WQs too.