I was looking for Kauffman & Sons Saddlery but alas it seems to be all gone. It use to be located on the corner of Lexington and 24th Street, then East 64th Street, then as a online entity... but now it's gone. However, I did find Manhattan Saddle, which is only half a block away from Kauffman's original store. The idea of visiting this store was to open my mind into a culture or sport that I've never had contact with. According to my thesis advisor, Kauffman's was a rarity especially since there aren't many riders actively riding in New York City that it'll make me think about its objects on display in a materiality sort of way. I think Manhattan Saddle did the trick.
Most of the objects that would come in contact with the horse like saddles and bridles were mainly leather-based. Most of the whips I found were of composite materials (saves weight?) Some stirrups and connection links were metallic.
In the helmet area, they had a sectioned display. Look at the assortment of layered materials that come together to make a helmet. There's definitely clear elements of intangible qualities in choosing materials for helmet design, not just for horse riding but for other purposes too; perhaps worth looking more into.