Type I infantry color disc. Pre 1937. S is for supply in one reference online. Not sure what the KV stands for. The bottom of those letters looks superimposed over the rifles.
US Militaria Forum has some knowledgeable people maybe post there also or I can post it for the OP there if needed. The KV has me perplexed, maybe ROTC or Military Prep School associated....?
I don’t think this has anything to do with snipers. Fourman110 is correct that it’s an enlisted man’s collar disc and that the ‘S’ letter where the unit or company designation is normally placed is for “Infantry Supply”.
It could be Kentucky Volunteers or Kansas volunteers as both states had volunteer infantry units in WW1. Kentucky had 153 field clerks. Not sure about Kansas.
The essential design is that of an enlisted man's collar disc, as used by US infantry in WWI and a little beyond. Quite how it connects to an Indian with the initials 'VKS' when the initials on the disc are 'KVS' escapes me.
These insignia come with various initials both above and below the rifles, for which the 'S' at the bottom is generally understood to be for a 'Supply' unit (ie Infantry Supply). The mystery is what the upper initials 'KV' might stand for.