Mirror Modifier's Apply button is grayed out even in Object Mode

It’s a linked duplicate (i.e. you have more than one object using the same mesh). Blender doesn’t allow to apply modifiers to linked duplicates.
To fix it manually you’ll need to make the object single-user, apply the modifier, then re-link to other objects. Or you can use the Modifier List add-on which can do that for you at the press of a button.