Lookup your child’s social security number over the phone? No.
This ehow article presents some false information:
Call the Social Security Administration at (800) 772-1213 and speak to a representative. It may be that you do not need the actual card at this time, because knowing the number is what’s important. The representative will be able to tell you your child’s Social Security number over the phone after verifying your identity. You will be asked for your Social Security number as well as for personal information.
http://www.ehow.com/how_5686728_child_s-social-security-number.html
You cannot call Social Security via phone and have them locate your child’s social security number. They will not do it.
Hopefully ehow will update their article soon to reflect this information!