You didn't say when your divorce was final. You can only file as married filing separately (MFS) for 2023 if you were still married on the last day of 2023. In other words, only if your divorce was not final until 2024. If the divorce was final on or before December 31, 2023, then you cannot file MFS for 2023, in which case you don't need her Social Security number. (That would also mean that the MFS return that your ex-wife filed is not valid).
Can you not ask your ex-wife for her Social Security number, explaining why you need it? If not, and if you don't have, and can't get, a copy of your 2022 joint tax return, another possible approach is to have your divorce lawyer contact her lawyer and ask her lawyer to provide you with her Social Security number, again explaining why you need it. You also need to know whether she claimed the standard deduction or itemized deductions on the 2023 MFS return that she filed, since, as xmasbaby0 pointed out, you have to do it the same way on your own MFS return.
