Is my wife eligible for a social security number?

First, congratulations on your third anniversary. Second, without the ability to ask clarifying questions, any response I give here will be inadequate, so please keep that in mind as you read this. Based on what I can glean from your statement, I think a few things are happening.  1. Your primary objective is to get health insurance for your wife. So that you know, your wife doesn't need an SSN to apply for private health insurance. She might need an SSN for government health insurance. You need to talk to a security disability benefits attorney to determine if she needs an SSN. 2. Your wife is a noncitizen who doesn't appear to have a Green Card. I recommend that you talk to an immigration attorney to help apply for your wife's green card. In the meantime, please visit the USCIS website to become familiar with the green card process. https://www.uscis.gov/green-card All that said, your question was about her eligibility for an SSN. The social security administration (SSA) is responsible for issuing SSNs. The SSA has different eligibility requirements for the two types of noncitizen-applicants (immigrant vs. nonimmigrants). Please refer to SSA's website or visit your local SSA office for up-to-date information. Here is a link to the SSA eligibility requirements for noncitizens: https://www.ssa.gov/pubs/EN-05-10096.pdf.If you meet the qualifications, you can apply online at their website (https://www.ssa.gov/ssnumber/) or visit your local office and apply in person. 

An immigration lawyer or family law attorney is in the best position to answer your question. They deal with social security and marital-related matters daily. Another option might be a social security disability benefits attorney. Even though your wife is not disabled, were you to die, you probably want to be sure she does run into even greater challenges if both of you assume she will receive survivor's benefits.