There is a difference between being in the country legally and having an authorization to work in the US, and having green card.
If you are not a US citizen, you are considered an alien. You can be a resident alien (also known as permanent resident or having a green card) or a non-resident alien (meaning you have some immigrant or non-immigrant status). There are quite a number of legal statuses in the latter category: a tourist, a student, a skilled worker, a dependent of skilled worker, a dependent of a diplomat, a spouse waiting for a green card, etc. Some statuses allow work (e.g., dependent of a diplomat), some allow limited work (e.g., a student can only work on campus, as a part of training, or by showing significant hardship), some don't allow any work (spouse of a skilled worker).
If your immigrant status doesn't allow work, you usually can get ITIN (tax identification number), and you potentially can get as SSN that says "Not valid for work", but you need to explain why you need SSN in that case.
If she is legal, you need to ask her what is her status. Form I-9 lists documents that she can produce to show that she is eligible to work in US. (PP is wrong -- unrestricted SS card is proof that you can work legally in the US, but you need is a photo ID like driver's license, as SS card doesn't have a photo).
What could be that she is present legally in the country, but doesn't have an authorization to work in whatever status she is in. You should ask her what her status is.
An alien is referred to as illegal alien if she/he doesn't have authorization to be physically present in the country (e.g., entered without inspection, overstayed visa, out of status).
Close, but not quite. If you produce a SS card for an I9, you need another document (birth certificate with seal) and a photo ID. I just had to do this. It has to do with proving both identity and legality to work.