How Do You Automatically Give Players An Item When They Kill Another User?

Firstly: You can put everything in the loop into a chain of command blocks: First repeating, pointing into the second one, which is chain, pointing into the next chain command block, etc. Set all of them to "always active" and it will always run. No need for redstone nowadays.

Did you look at the output of your command blocks? They don't print errors in chat, you have to open them to see their result. You have a syntax error, the /give command is incomplete, it lacks the number of items. Change it to give @a[score_KillScore_min=1] minecraft:emerald 1

Edit: The count is not actually required. This should not have fixed it, but according to the asker, it did. I don't know why.