You can't claim a dependent without either an SSN or ITIN. Typically, what you do here is print your return and file by mail. Include a form W-7 ITIN application and all required proofs according to the W-7 instructions, and mail everything to the IRS address for ITIN applications. After they issue the ITIN and return your proofs, they will process the tax return. You will file your state tax return by mail after getting the ITIN.
If your child is eligible for a social security number, you should never apply for an ITIN. Apply for the SSN and don't file your tax return until the SSN arrives, even if it means filing a late tax return. You can get an automatic deadline extension to October 15.
