If a child is under age, whether there is a will or not, they have to go to probate or court and other proceedings to assign a guardian. This would be what should be in the White County, IL court records.
If a man left underage children, you should expect to find a guardian being appointed and the children being referred to as "orphans"
I ran across the second marriage record for James Enoch Walton to Mrs. Elizabeth (nee Tucker) Trusty. His parents are listed as Nimrod Walton and Susan S. Pears