Attempting to get an allay to pick up armour will result in the allay failing to do so if the damage value of the armour they are trying to pick up is not same as the damage value of the armour they are holding.
Example: if an allay is carrying an iron helmet with 50% durability left it will not be able to pick up any iron helmets with any other durability percentage.
This appears to be a problem with armour only as they are able to pick up tools regardless of the differences between the damage of the item on the ground and in their hand.
Please confirm your game version. You can see it on the main screen in the lower right corner.