State tax filing

Well, its first important to note that WI and IL are reciprocal tax states....and as such, an IL resident either physically working a W-2 job in WI  , or even if some is done remotely from home in IL, only has to pay IL taxes on that W-2 income.  (you may already know this)

Problem is, that not all WI companies are registered with IL in a way that allows them to withhold IL taxes for you with each paycheck.  This sounds like what your WI employer is telling you.   (Many WI employers can withhold IL taxes for their workers who cross the border from IL to work for them...but not all can/do)

This being your situation, you are expected to pay your "estimated" IL taxes yourself on a quarterly basis during the year using the IL-1040-ES  payment forms  IF...IF...your Balance Due for IL taxes for at tax time are expected to exceed $1000, if you did not pay those estimated taxes....These payments are "generally" due to be paid by the 15th of April, June, September of 2022, and January of 2023  (But because of weekend dates, two of the dates are extended 18 Apr 2022, and 17 Jan 2023 ).

You should read the IL details at:

Individual Income Tax Estimated Payments - Filing Requirements (illinois.gov)

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There is a link to the IL-1040-ES tax forms on that page....but you may be stuck making a large ES payment for the full tax year as we are now in the last quarter.  Doing so might reduce any penalties or interest for underpayment that might have accrued.

There are some exceptions for having the penalties/interest applied if you wait until tax time to file&pay...but that really depends on the specific details of each person's prior year IL tax situation...how much the IL tax liability was last year (for 2021 taxes)...and what the IL tax liability ends up being this year.  (Tax liability is the IL tax total assessed Before considering any IL withholding or IL estimated tax payments done during the year)