If you have withholding for a state, you can file a state return to request a refund. You will not need the state ID for your federal taxes, however the withholding in box 17 will need to flow from the W-2 entry in the federal return to the state return. Try using the federal ID in box 15, or if the state requires a particular format, such as ending in XYZ, then enter 9999999XYZ for the number of digits it specifies.
Your state department of revenue will be able to easily identify your employer from the 9-digit federal EIN, so the state ID number is a little redundant anyway, but is still a way that your state tracks employers in their internal state computer system. The state employer ID number is a bureaucratic convenience for your state's department of revenue; it is not an essential element to the correct filing and processing of a valid personal income tax return (either state or federal)
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