My family and I live in Germany. I'm German and my wife is American (SSN holder), while our two children have dual citizenship, both have a CRBA but both don't have an SSN. Can my wife claim them as dependents in the IRS stimulus payment?
In order to claim a dependent, you must enter their SSN or ITIN on your tax return. Since your children are US citizens, they are eligible for SSNs, and thus cannot get ITINs. Therefore, you must get SSNs for them in order to claim them as dependents.
In order for you to children to qualify for the $500 stimulus amount and the Child Tax Credit, their Social Security Numbers must have been issued before the due date of the tax return. If you get their SSNs after the due date of the tax return, even if you file your tax return late, or amend your previously filed tax return to claim them as dependents, they cannot get the Child Tax Credit (though they can get the $500 credit for other dependents), and they cannot get the stimulus check.
If you cannot get the stimulus "check" amount for the children because you don't claim them on your 2019 tax return, as long as you get them SSNs before the due date of the 2020 tax return (in 2021), you can claim them as dependents on your 2020 tax return and get the part of the stimulus payment for your children, since the stimulus payment you qualify for 2020 exceeds the amount you got in the check by that much.