Can I go ahead and file my return without having my kids social security numbers

you won't be able to e-File.

Try to avoid sending a paper tax return with incomplete information.

EVERYONE listed on a tax return must have a SS#, ITIN or ATIN.   Since the IRS will not start processing returns until 2/12 you have the time to look up those numbers.  

If you are waiting for SS# to be issued then you cannot file until it is issued ... go on extension if needed but do NOT file without it.   If you file before the number is issued you cannot amend later to get any of the child related credits ... they will be forfeited.