I'm not familiar with Chippewas. Were they in New York? I see from a Google search that there are Chippewa reservations in Michigan and going west to Montana. They are also in Canada. So if Winnie visited the reservation it would have been in those places, not Illinois or Oklahoma.
Winnie's lineage goes back to New York on both her father's and mother's sides.
On April 3, 1832, Peter Striker married Harriet Robins in Trumbull Co, OH
1850 United States Federal Census Springfield, LaGrange, Indiana Peter Striker 49m NY shoemaker Hariet Striker 38 f NY Maria Striker 17 f OH Simeon Striker 15 m OH Amanda Striker 13 f IN Charlotte Striker 10 f OH James Striker 8 m IN Almira Striker 7 f IN ------> Winnie's mother Albert Striker 1 m IN
1860 United States Federal Census Ridgeville, Monroe, Wisconsin Peter Stryker 57 m NY farmer Harriet Stryker 47 f NY Jas M Stryker 18 m IN Albert E Stryker 11 m IN F E Stryker 6 m IN (living next door) Henry Lincoln 28 m NY farmer Almira Lincoln 16 f IN David Gamble 23 m NY Frances Gamble 17 f NY Almeda Gamble 1 f WI
1870 United States Federal Census Nunda, McHenry, Illinois John Henderson 27 m w IL farmer Almira Henderson 25 f w IN keeping house Harriett Lincoln 9 f w WI Henrietta Lincoln 5 f w IL William Fuller 15 m w IL Adelia Cotting 29 f w IL Emma Schmidt 2 f w IL
There's information on the Striker line on World Connect: