Trying to claim exempt in Illinois state. At what point is this allowed

Yes.  IL and IA have a tax reciprocity agreement which means your wages/salary should only be taxed by your resident state (IA).  To get a refund of the incorrectly withheld IL taxes follow these steps.

1. First click the Federal "My Info"  section
2. "Edit" your information
3. Be sure you indicate you lived in IA on 31 Dec and did not live in IL in 2023 (IL is not selected as resident state)
4. Continue
5. Scroll down to "Other State Income"
6. Answer "Yes" you earned income in another state
7. Select IL (this will generate an IL non-resident return)
8. Continue
9. State Taxes in the left hand menu
10. Prepare State Taxes
11. Continue the IL interview
12. "Edit" the "General" category
13. Ensure you are shown as non-resident if not, change it
14. Did you earn money in IL but  never live there "Yes"
15. Select Resident of IA
16. Continue
17. "Edit" the "Income" section
18. Verify no other IL income items apply "Continue"
19. Next page should be your "Wages"
20. Enter "0" (zero) in the box for IL wages
21. Continue and enter zero in IL boxes for other incomes as needed.
22. Your IL refund at the top of the page should now be what was withheld for IL

Additionally, your employer should only withhold for your resident state.  If you haven't already you need to contact your employer to stop IL withholding and start IA withholding going forward.