You will need to check with your state to see if they accept state tax return filings with W-7s attached
State filing - You will want to file an extension for your state until you receive your ITIN. This is because the application for your ITIN is a federal filing. Most state will not process your return without either a SSN or an ITIN. (However some states will accept state filings with W-7 attachments. You will need to contact your state's Department of Revenue and ask if your state will accept a state tax return with a Form W-7 attached).
When the IRS assigns you an ITIN, update your returns (both federal and state) in TurboTax with this new information and then you should be able to e-file your state return.
Because many states give an automatic extension when you file the federal extension Form 4868, just file a federal extension even if you don't think you will need it for your federal tax return. But check with your state's Department of Revenue to make sure that your state doesn't have a separate state extension filing. Here is a link to all state's DoR: (to expand answer, select see entire answer)
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