I can't seem to figure it out.
I went looking in the game folder for a manual to explain it... but I can't seem to find one.

Assuming you have ammo storage and have bought ammo at the shipyard for the correct weapon types, the gunner should reload the weapon they are assigned to when needed during a battle.

That's strange.

I have two turrets on my ship, with a full ammo storage right next to them. But my gunners just sit in the guns and their combat icons just say no ammo. But I have more ammo available (I can see the ammo in the storage).

Not sure why they're not moving.

Make sure it's the correct ammo type as zogstrom noted. For turrets, you want bullets, not torpedos or grenades (those are for torpedo launchers and cannons respectively.)