I'm not a lawyer, so this isn't legal advice. For that, you need a lawyer. However . . .
If her reluctance to provide a social security number is either some misguided attempt at privacy or a resentment of government somehow, then ask your lawyer about this approach. Have that sibling quit claim her interest in the property to one of the other siblings. Then she's no longer one of the owners. So now there are only seven owners, all of whom are willing to provide their social security numbers.
As a side agreement, your siblings can agree to compensate the reluctant sibling for her portion that she gave up.
Example: Suppose the property is worth $56,000. If that were divided 8 ways, you'd each get $7,000. The reluctant one gives up her share. Now the $56,000 is divided 7 ways, so you'd each get $8,000. Then each of the siblings gives $1,000 back to the reluctant one, and you each end up with $7,000 . . . just as you would have at the beginning.
In addition, check with an accountant to make sure that it's done properly from a tax standpoint.
Hope that helps. [You've already given someone else a "Best Answer," but maybe my answer deserves at least a "Thumbs Up"?]