Do I need an EAD to get an SSN? Am on L2 visa.

Yes, because the SSN is not issued unless you have work authorization. The L-2 by itself does not convey work authorization. You must apply for an EAD.

A person who is on L-2 status should apply to the US CIS for an Employment Authorization Document (EAD) as soon as possible. File it with proof of your L-2 status (and filing fee). Once the EAD is issued, then apply for a Social Security card. It will be granted.