If you are within the U.S., you should go to a Social Security Administration office to apply for the SSNs for your children. File an extension while you are waiting for the SSNs.
I am not within the US. My children have never visited the US. That was not my question.
You can not e-file with fake social security numbers. The IRS uses Social Security numbers or ITINs to identify taxpayers and dependents. Returns can be rejected when there is a mismatch between the SSN and name. Go to a close U.S. Embassy to apply for their SSNs.
You may also consider removing your children from your tax return so that you can e-file.
Are you suggesting that all US citizens are legally required to petition the IRS to obtain SSNs?
Yes, if your children are US citizens they should have a social security number. You need to go to the Social Security Administration and apply for social security card for each of your children.
You do not petition the IRS for a social security number. Social Security numbers are obtained by completing an application with the Social Security Administration. You can do this online by going to Social Security Administration.