That would depend on whether or not you currently have a Verizon line of service.
If you do not have a Verizon line of service, then you need SSN to activate a new account on Verizon.
As tikibar1 has said, you can get free SIM cards from your local Verizon corporate store. You can also have one sent to you for free by calling CS.
Since you are posting in the Samsung Galaxy Note 3 space, you may want to make sure you get one of the NFC SIM cards in case you were to want to use ISIS on your phone.