Yes, you can e-file the Illinois state tax return using TurboTax.
You can either e-file the federal and state tax returns at the same time or you can e-file the federal return before the state return. If e-filing the state return after the federal was e-filed you must wait for the federal tax return to be accepted by the IRS before you can e-file the state tax return.
You cannot e-file the state tax return before the federal return is e-filed.