After you file

Intuit can't to do that.   Any information that you enter in the TurboTax program as a W-2 could be faked or altered, even if it is first directly imported from the provider.  Therefore, the TurboTax W-2 information worksheets are not legal proof of income and are not legally testable copies of your actual W-2. 

(In New York State for example, TurboTax prepares a substitute W-2 form as part of the state tax return package. This form contains all the information from your W-2s, and is sent to the state in lieu of actual W-2s. But, the state then matches the information against federal records, and will send a notice if the information does not match.)

If Illinois requires a physical copy of your authentic W-2. That the only way to provide that is to mail it to them. This would mean either, mailing your entire tax return, or even filing your tax return and then mailing the W-2s with a cover letter.  If the cover letter procedure is used by Illinois, TurboTax will prompt you to print out the necessary cover letter, and you will have to put it in an envelope with a stamp and your W-2s.