Unfortunately, there is no way to claim a child as a Dependent on your tax return without their social security number.
Please review the qualifications you must meet in order to claim a child as a Dependent here. If you meet all of the qualifications, I suggest you contact the county to see if they will give you the SSN.
The IRS also has a tool that you can use to help you determine if you qualify to claim someone as a Dependent: Whom may I claim as a Dependent?
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