It's too late to change your W-2s for 2023 or earlier. For 2024, even though your employer has been withholding for the wrong state all year, they could still correct it for this year, so that your W-2 would be correct. They might give you a hard time about doing it, or say that they can't do it. They might even try to convince you that what they are doing is correct.
Here's what should be happening. Since you work in Illinois, they should be withholding Illinois state income tax and not Ohio state tax. Your W-2 should have a line of state information with IL in box 15, your total wages in box 16, and the Illinois tax that was withheld in box 17. You file an Ohio resident tax return and an Illinois nonresident tax return. Both state tax returns will show tax on your Illinois income, but you will get a credit on your Ohio tax return for part or all of the tax that you pay to Illinois.
I don't know how the Ohio school district tax works. @Hal_Al will be able to explain that. You might have to have a separate OH line on your W-2 for just the local tax.
From the way you describe the expense reimbursement, it sounds like it might be an accountable plan. Ask the accounting department specifically if that's what it is. If so, the expense reimbursements should not be included in the amounts on your W-2, and you don't put anything on your tax return about the expenses.
If you want to go back and correct earlier years, for each year you would have to file an amended Ohio tax return and a new Illinois nonresident tax return. You would get a big refund from Ohio, and have to make a large payment to Illinois, plus penalties and interest for filing and paying late. You can probably do this for 2021, 2022, and 2023. The problem you would have is that your W-2s show all your income as Ohio income. If you file an amended Ohio return saying that it's not Ohio income, Ohio is going to say you're wrong, and they won't pay the refund. To prove that it's Illinois income you would probably need a letter from your employer saying that you worked in Illinois and the W-2s are wrong, or a corrected W-2 for each year showing the Illinois wages and zero in box 16 for Ohio. I expect that you would have trouble getting that kind of proof from your employer. Since Illinois hasn't challenged it (yet), it might just be better to let sleeping dogs lie, even though it's wrong, but do it correctly going forward. But legally, you should correct the past years. You might want to ask a local tax professional for help in dealing with this.
