Illinois has pretty strict ID requirements for electronically filing tax returns. I have not seen any news from the state that indicates they have re-programmed their tax systems to honor the same three-month extension.
You could give it a try and if the return will not e-file you would still be able to e-file your federal tax return and you could either print and mail your Illinois return or you could save it in hopes that your state's DMV re-opens soon so that you could transmit that return.
@Hersheydog5
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