Looking at the different levels of modship Gaia has, they definitely don't require that for every mod position. Some mod positions are as basic as forum mods, which wouldn't require an SSN any more than you had to supply one to get your position.

I wouldn't be surprised if they required an SSN to be given a mod position that has unbanning privileges, but at the same time any payroll position would require an SSN, and they may also only give banning permissions to mods on payroll as incentive not to abuse it.