A William Wallace served as 2nd Corporal in Captain Alexander White's Company of Mounted Volunteers, according to the muster roll. He enrolled May 26th, 1832 and was discharged June 15th, 1832.
This is the same company that Abraham Lincoln served in during the Black Hawk War.
You might try visiting the Illinois State Archives at http://cyberdriveillinois.com/departments/archives/databases.htmlhttp://cyberdriveillinois.com/departments/archives/databases.html and search their database of Illinois Volunteers of the Black Hawk War. You can submit a request via email or post and they will look-up your ancestor's name in the book "The Black Hawk War 1831-1832" by Ellen M. Whitney. The book doesn't provide any real genealogical information, other than confirming your ancestor's service.
I submitted a request and received 4 or 5 pages in about 3 weeks time.