SSA will likely not issue her an SSN so soon to I94 expiry.   Again, it doesn't matter if it's the exchange, a private plan or whatever. An SSN is not required by law, the same law in fact that created the ACA. 😉 It's likely you'll need to speak with someone on the phone, as internet web forms can be a one size fits all approach and does not handle more unique situations. Hundreds of couples on here have signed up without it, and you can explore those threads. But, it is dealing with t