I believe that since militias were local, you may want to begin at the Indiana Archives and Records Administration. I am sure someone from A2 will provide a follow-up answer regarding NARA holdings. In the meantime, read this PDF from the National Archives. It may provide some additional information you may need. https://www.archives.gov/files/research/military/indian-wars/black-hawk-war-1831-1832.pdf
Dear Ms. Loveless,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located 8 series pertaining to the Black Hawk War to consider for your research. Please contact the National Archives at Washington - Textual Reference (RDT1) via email to inquire about John Dawson and the Brenton County Militia in these records.
We hope this is helpful. Best of luck with your family research!