That shouldn't really pose a problem; or at least not one that will prevent you from filing your tax return. Please allow me to explain.
The only truly meaningful 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. If there was a mistake made with any of these items, then that would definitely be something that requires attention.
However, with respect to the state number ID issue, it won't really matter what you input in Box 15 Employer ID, as long as the software will accept it (explained below). Simply try leaving the space entirely blank at first. If there is any issue with Box 15 that prevents e-filing your return with a blank Box 15, and it absolutely requires some value in the software, then either try using your Federal EIN here, or enter a "nonsense" 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, for instance, then enter a just enter a value that conforms to that format . . . such as 123456XYZ. This will enable you to bypass the issue and file your return. In other words, try to be a little creative here, and just give the software some value (even gibberish) that it will accept. This can include also using the Box 15 drop-down menu to select your state, if needed.
As you correctly point out, some states don't have an income tax, or tax out income taxes by withholding.
In fact, the IRS computers and your state department of revenue (if applicable) really won't care what you do here in the TurboTax program, as, they're really just looking for the federal EIN, SSN, and relevant wage data. The State ID is simply a bureaucratic convention for your state; the state 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, as long as you can satisfy the software with some acceptable input (or none at all).
Once again, your state should still be able to understand and process your tax return. As a very last resort, if nothing else works, you attempt to e-file and your state return is still rejected, then you can always ask your employer payroll office or human resources department for their actual number. If your employer has employees in your state (they must, because they employ you), then they will have a state tax ID number somewhere in their files, even if it's not printed on your W-2.Thank you for asking this question.