The state return will need to be amended after the ITIN is issued. You should be able to complete this in TurboTax. Presently, the IRS is taking around 4-6 weeks to process ITIN applications.
The IRS places this number for you on federal returns but, of course, they are not involved in your state return.
Since your return was filed without this information, I would recommend coming back in a few weeks after the ITIN is received to amend the return to add this information.
In the meantime, if you receive any correspondence from the state, reply to let them know you are waiting for an ITIN to be issued.
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