You can absolutely file your tax return without the GA Employer ID number, but the program may not let you e-file.
Here is a discussion from @GeoffreyG that covers this issue and offers some simple work-arounds:
"If you have no separate state business identification number on your W-2 form, then just try using your employer's federal employer identification number (EIN) again in Box 15. That will look basically look something like the following format: 73-1234567. Your state should be able to match your W-2 to your employer's reported wage information using the federal number (if necessary).
This should then allow you to e-file and get around any TurboTax program "error" messages, by inputting something (a value) into the field. If you try that approach but still encounter problems, then try entering some variation on a "nonsense" value such as 55555 55555 (with a space between), or a series of consecutive zeros or integers.
If you want to go to the trouble, you could also just wait and ask your employer's payroll or human resources office. They should have this information on file.
In any event, the IRS computers, and your state department of revenue, really should not care, as they're really just looking for the Federal EIN, employee Social Security Number, and relevant wage data. The State ID is mainly a bureaucratic convenience for your state. In other words, this single issue should not affect the processing of your tax return.
If this blockage prevents you from e-filing, you can't get around it, and you cannot obtain a state ID number from your employer, then you always have the "back up" option of paper-filing a traditional tax return.
