You really do not need this number to file your tax return; but please allow us to explain.
While it is preferred that you use an employer's state ID number if you have one, it's not essential if you don't. The only truly indispensable elements on a W-2 for income tax filing purposes are your employer's federal EIN (Employer's ID number), your own Social Security Number (SSN), and your actual wage data.
In fact, the IRS computers and your state department of revenue really won't need this number, as, again, they're really just looking for the EIN, SSN, and relevant wage data. The State ID is simply a bureaucratic convention for your state; they can and will still be able to easily and uniquely identify your employer by their federal EIN, even if they have to cross-reference the Federal / State ID numbers in their database. In other words, this single issue should not affect the processing of your tax return.
Also, if you are employed and living in a state without an income tax, try removing the wage data in Box 16.
If there is any issue with Box 15 that prevents e-filing your return, and it absolutely requires a value in the software, then either use your Federal EIN, or enter a "fictitious" value such as 12345 67890 (with a space in the middle, or not). If the TurboTax program asks you for a state ID number in a specific format, such as needing exactly 9-digits and ending in XYZ, then enter a value that conforms to that format . . . such as 123456XYZ. This should enable you to bypass the issue and file your tax return normally.
Thank you for asking this question.