Hello. In morrowind(without mods), when enchanting a constant effect of restore fatigue on a shield, does the restore fatigue constant effect work even if you're not in battle stance?

The enchantment will only work when the shield is equipped regardless if the player is in a fight or not.

You mean regardless if the player is in battle stance or not?

The term "Battle stance" is not specifically used in Morrowind, although "in combat" means about the same, in that the character is considered to be "fighting", and "battle music" begins. That can happen if any NPC or creature which is hostile toward the player is within some distance of the player character.

Shields can be equipped at any time, occupy a hand slot, and are not specifically held at the ready, regardless of whether you're in combat or not. In the event that an attack is made against the character, there is a random chance, modified by the character's Shield skill, that the character will be able to position an equipped shield in the way of the attack. It only matters that the character is carrying the shield in their hand, not slinging it across their back or otherwise storing it out of the way.

As long as the shield is equipped (carried in one hand), any Constant Effect enchantments will be active. Combat has nothing to do with it.

Anything "equipped" in the inventory screen, with that embossed border around the icon, will convey its effects. They do not have to be visible or unsheathed!