If one would buy Icebourn Gauntlet and Black Cleaver, would everyone's armor get reduced that got hit by the Black Cleaver? I know it works for the Spirit of the Elder Lizard because there is a clear particle animation. I basically want to know this for Ezreal, but I guess it goes for every champion.
Yes every champion that suffers damage from the AOE effect will take a single stack. Note however that there will be only one stack applied to the main target that procs the on-hit effect. This works similar to the on-hit of Ravenous Hydra or Tiamat (Source).
As the description of black cleaver says it will apply a stack for every time you deal Physical damage. This counts for Items, Spells and auto attack but it does not apply for on hit effects on a single target. However spells that hit multiple times in a short amount of time will apply a stack for every single time they hit a champion. This is the main reason why BC is so popular on champions like Wukong (Ultimate) and Pantheon (E-Spell).