Does both rider and mount take full damage from a lightning arrow if it hits?
Lightning Arrow says that:
Make the attack roll as normal. The target takes 4d8 lightning damage on a hit, or half as much damage on a miss, instead of the weapon's normal damage.
Whether you hit or miss, each creature within 10 feet of the target must make a Dexterity saving throw. Each of these creatures takes 2d8 lightning damage on a failed save, or half as much damage on a successful one.
So, first, you make a normal attack roll. That means that if you shoot it at a mounted character, you'll need to shoot at either the rider or the mount, since attack rolls are made against a single creature. That creature, whether it's the rider or the mount, will take 4d8 damage (or half that if you miss). Then, each creature within 10 feet of the target gets hit by 2d8 damage, or half that if they make their saving throw. Note that this does not include the target.
Just to make it really clear: If you shoot at the rider, the rider takes up to 4d8 damage and the mount takes up to 2d8 damage. If you shoot at the mount, the mount takes up to 4d8 damage and the rider takes up to 2d8 damage.