I was stuck on this one for a while. I actually forfeited the quest because I couldn't get it to work.
I came back today, and basically started this whole quest line over, and worked on my Terrmian earring 1k kills. I specifically followed points on map, and killed the Saunils laying around, and then went to the furthest point, which popped up a dialog box and told me to place what I had found.
I guess the confusion was in that there isn't an actual material item in your storage. But also I couldn't get it to work, no matter how many Saunils I killed last time. Not sure if that is because I logged out and back in or what.
The next mission is much like this, but you kill Rhutums, and it actually gives you a count of how many shamans you get the spears from which is 10.
Good luck.

Where to put the armor piece of Saunil Guard near Rhutum District. You should not follow where the game tells you to go, it's the complete opposite... I hope it will help players who struggle to find the correct spot
Check out the screenshots section

09-2022: for me the armor needs to be handed over on the exact spot of the quest marker. EU PC game.