I remember a question along these lines came up before. The general consensuses was that there aren't any specific rules saying if it can or can't. But that having the animal companion learn a trick to active it was a reasonable way to handle it.
That being said, mounts are inherently smarter then animal companions and it could be argued that it should be capable of making sounds that could be turned into a command word that would activate the armor and be smart enough to understand that making that sound activates the armor.
Based on the rules, an animal companion can't activate a command word unless it can speak (unlikely) and there are no tricks that allow a substitute. I certainly wouldn't expect to be able to do this in an environment like PFS.
For a home game, a GM allowing a custom animal friendly command word and a trick or an item that the PC rather than the animal companion activates wouldn't be unreasonable.