Yes, the judge may deny your request if he deems that you don't meet the requirements of the statute for "primary caregiver" and "undue hardship". 705 ILCS 305/10.2 (b).
You should examine your jury summons carefully for contact information. You may find it possible to obtain an excuse; in some counties a "free pass" or two is allowable--but you'd have to request it in sufficient time. There are counties where you make a call the night before to find out whether you will be required to report. You may also find out whether there is a child care service connected with the courthouse. If you are in Pike County, you may find answers to these questions from the Circuit Clerk 217-285-6612.
The information provided here should not be construed to be formal legal advice. The provision of this general advice does not create a lawyer-client relationship. Persons with legal questions are encouraged to seek independent counsel for advice regarding their individual legal issues.