Wights
These are the undead minions of the White Walkers turned by them for their armies. If you want a common analogy for them they resemble zombies as in they are slow and have their power in numbers and regular means not being able to kill them. If you watch closely most of their kills are actually due to swarming a target where they claw, bite and maul it. They are not adapt fighters. As such it is very unlikely they would be able to use a bow.
White Walkers
The commanders of the undead army, they seem very intelligent with thoughts of their own and are likely a species of their own. However, their weapons are actually made of ice and even though they are very good at manipulating the material I doubt they'll be able to form it into a bow string.
With that said we do see an example of the Night King using a ranged weapon, albeit a spear. He is also very accurate with it considering what he does with it too.
As for why they didn't use spears to kill the merry men? Well why waste resources on a group of people who are going to die anyway. They were surrounded and freezing to death with no food or supplies, it's likely the undead were just waiting for the inevitable.
There is however, the theory that the Night King lured the group North to get a dragon to come up so he could kill it and turn it. I personally do not buy into this but it is certainly possible until confirmed either way.