Your question is unclear; are you asking whether after the I-797 (Approval Notice) for the adjustment application you can apply for a social security card? If you've been approved for adjustment but have not yet received your actual card then wait for the card. Usually they have been taking about 2 weeks to get the card after the approval. This is actually very fast and CIS deserves praise for the speediness of getting people their green cards after approval. Used to be much longer.
No attorney-client relationship is created or implied by this communication in any way. Consult a competent immigration attorney preferably one who is a member of the American Immigration Lawyers Association (AILA).
Your I-797 is simply to notify you of the status of your application or petition. Once you receive your actual green card, it can be used to apply for your Social Security Card and a state issued driver’s license. It can also be used to prove employment eligibility in the u.s.
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