If my child made less than $5k can we file a refund for the state taxes paid to Illinois?

Yes, you can file for a federal and state refund if there was withholding on their w-2.   Create an account and file a return for your child, but make sure you mark the box that says Someone else can/will claim me.   You can still claim your child on your return, but make sure to mark that box on their return, so your return does not get rejected.   Do not add the income to your return.  

as a dependent, for Illinois there probably is no personal exemption, so the child will be taxed at 4.95% on the Illinois income. Any excess withholdings will be refunded. 

more specifically, if the child is a dependent and has base income (line 9) of more than $2426 the exemption is zero.