If your wife does not have a SSN or ITIN, she should make an application for at least one or the other. This would have nothing to do with you and your withholding tax, unless your spouse is a non-resident alien and have an impact on your married filing joint tax return. You cannot file as Single, only Married or Married Filing Separate.
https://ttlc.intuit.com/community/taxes/discussion/i-m-married-but-my-wife-doesn-t-have-ssn-yet-so-i...
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