Can I still receive a child care tax break if I am paying my relative and they do not claim my payment as income on their taxes?

Well you have to enter her ssn on your return.  Then the IRS will probably ask her if she doesn't report it.  

In order to claim the Child and dependent care tax credit, you have to enter the caregiver's SSN on your tax return.

When you do this, the IRS knows that your sister has been paid the amount that you claim.

If she does not report this income, she might get a letter from the IRS.

Whether or not the caregiver reports their income is not your problem.  You report the amount paid, caregiver name, address and Social Security number of the caregiver on your tax return.