Do settlers use ammo I store in an "armory" in Fallout 4?

Settlers only need to have one of any given ammo in their inventories, in order to use a weapon infinitely. They do not run out of ammo UNLESS they are using big guns, like a missile launcher or mini-nuke. 

Yes, settlers (and provisioners) will use weapons left on weapon racks or stored in the settlement. I've personally had this happen MANY times in my settlements, where I will put a nice legendary weapon on display, or store some stuff in a chest to use later, it goes missing, and then I find it in a settler's inventory. 

That said, they seem to have preferences. My settlers seem to like laser guns and ballistic rifles, even if weapons with better stats are available. 

You do know that you can give the settlers weapons, armor and ammo, right? No need to leave it in a chest for them to grab. If you check their inventories and they already have weapons, that could also be why they don't bother grabbing the stuff that you've left for them. They will pick up weapons and ammo from fallen enemies, so they may have already grabbed things from raiders, gunners, or mutants. 

I sometimes leave suits of power armor, with fusion cores inside, for settlers to use. Sometimes they will use it, sometimes they don't. I don't know what makes a settler decide to use something or not. May have to do with proximity to the weapon or power armor and threat level from the attack. 

Possibly, but I'm kinda thinking they don't. They may be more inclined to use any ammo they have on their person, and if they run out of that they might run in bare handed...unless they find something else nearby 🤔