Legally, she is correct. One generally doesn't have to voluntarily provide their SSN to a non-governmental body.
A court, can order the home sold and the proceeds divided between the eight siblings.
If one doesn't assist in the sale willingly (failure to provide her SSN), the court could order her to comply.
Between her refusal and a judge's eventual order, much time can pass and money will change hands.
I'd try and reason with her, cajole her, even incentivize her; that shortens the time and reduces the costs.
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