Can I file an income tax return electronically without an employer's sate ID number?

It depends.

Do you live in a state that does not assess individual income taxes?  If so, you should not have any amounts listed in boxes 15-17.  If that is the case, make sure all of the boxes are empty in TurboTax.  This should clear the error and allow you to electronically file your return.

If however, you reside in a state that assesses individual income taxes, boxes 15-17 should be completed.  You would need to contact your employer to confirm that your W-2 is correct and possibly request a corrected form to ensure your amounts withheld are reported to the state taxing agency.