Yes, you can fire it as a reaction
The property states:
[Y]ou can fire only one piece of ammunition from it when you use an action, bonus action, or reaction to fire it, regardless of the number of attacks you can normally make.
It enumerates each type of action you can use to make an attack with. It does not say:
[Y]ou can fire only one piece of ammunition from it once per round, regardless of the number of attacks you can normally make.
That is, you can fire one piece of ammunition from it when you use an action. You can also only fire one piece of ammunition when you use a bonus action. And you can only fire one piece of ammunition from it when you use a reaction. This is true regardless of the number of attacks you can normally make.
What that property does not say is that you can only make one attack from that same weapon once per round.
The key here is that they limited the loading property to one attack per action.
Application: War Priest
The War Cleric's ability War Priest lets them use a bonus action to make a weapon attack when they take the Attack action on their turn. So they can fire from a crossbow with their action, and they can fire from it again using their bonus action. If someone they were in melee with tried running without disengaging first, they can make another attack using a reaction.