Most designs of Indian saddles had stirrups.
"The Indian stirrups designed for everyday use were quite simple. Cottonwood or poplar branches were bent in a U shape and attached to a flat wooden base." – Thomas E. Mails, Mystic Warrior of the Plains"
They also often traded for metal stirrups with the Spanish and Americans. However, most warriors did not use the saddle when riding into battle:
"Everyone used saddles for general travel because other items could be hung on them fore and aft, but men usually removed theirs and rode bareback or on a robe when hunting or going into battle." – Thomas E. Mails, "Mystic Warrior of the Plains"
Nick is partially right in that American Indians rode bareback in battle, but used saddles for everyday use.
I think the vignette is misnamed. Instead of "War Party", it should probably be labeled "War Pony." There is no "party" of individuals. The horse or pony is the main focus of the vignette and is probably better suited describing daily life than warfare.