No. You need the children's SSN to claim them as dependents. He could file for an extension which would give him until October 15. The form to file for an extension will be available in a week or two. It is a form 4868. Meanwhile--not sure why you think it would take so long to get the SSN's.
To Replace a Child’s Social Security card
https://faq.ssa.gov/en-US/Topic/article/KA-02723
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