US citizens don't use a W-7 or need an ITIN.  They use SS-5 to apply for an SSN.

Question 4: Where can a U.S. citizen overseas apply for a Social Security card?

Answer: U.S. citizens living abroad may apply for a first-time Social Security card or a replacement Social Security card at any Social Security Federal Benefits Unit (FBU) at designated U.S. embassies or
consulates. The Social Security Administration (SSA) maintains a list of FBUs at  www.ssa.gov/foreign/foreign.htm.
An applicant for an original SSN who is under age 12 may apply by mail. The applicant can send the completed Social Security form SS-5 and required proofs (must be original documents or copies certified by the custodian of the record) to the servicing FBU. Applicants age 12 and older who have never had a SSN before must have an in-person interview.

https://www.ssa.gov/foreign/joint_frequently_asked_questions.pdf