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No, you can't get a W-2 form from the state, the only place you can get a W-2 form is from your employer. 

If your employer will not provide you with your W-2 form, you should call the IRS at 1-800-829-1040 and they will contact your employer. You will need basic information such as your and your employer's address, phone number and dates worked. If you don't get the W-2 by the due date of your tax return you can file for an automatic six month extension of time to file your federal tax return.

You can learn more here: What to do if can't get W-2 form

If you can't get the W-2 form, you can create a substitute W-2 Form in TurboTax. You do that by entering your W-2 information in that section of TurboTax, if you know the basic information such as earnings and taxes withheld. After you enter the information, you will see an option to indicate that you need to prepare a substitute W-2 form 4852. You will have to print that form and mail with your tax return.

You can read more on that process here:  How do I file a substitute W-2 using Form 4852? (intuit.com)