There is no public source for your employer's state ID. It should be on the tax form. On a W-2, it would be next to the state abbreviation on line 15. However, if your state does not have an income tax, then boxes 15 through 20 can be left blank. If you need it, contact the employer.
Hopefully you're not trying to file based on pay stubs instead of a W-2. If so, that's a no-no. The employer is supposed to provide you a W-2 by January 31.
If you have your W-2 and there is no State Employer ID shown, or TurboTax does not like that number (be sure you entered it correctly), you can try using the employer's Federal ID for that entry.
Here's more info on Employer State ID Number.