Unfortunately, you have to have a social security number for the child to be claimed on your taxes.
I have found the SSN for clients in several ways but it will require you to have the appropriate paperwork.
In my state, you need a SSN number to have Medicaid, for example. In many states, the SSN functions as their individual ID number in the foster care system - though many case workers don't know that.
The SS numbers are usually filed for by the hospital staff when the babies are born - right along with filling out the birth certificate.
If you have anything that lists their full legal names, you can usually take it down to the SS office with your paperwork and ask what the numbers are - if they can find the kids in their system, they'll tell you the numbers.
A word of caution, if the child was reunified, the parent may have used the child as a deduction already. If you do find the number and claim the child and it is a duplicate, your return will be held up and the IRS will most likely give the deduction to the parent. Sorry.