Vera Parra, I organize with immigrant youth in New Jersey
I am curious which state you live in and what schools you have looked at, that say the wont accept undocumented students outright. South Carolina and Alabama are the only states with laws that specifically ban undocumented youth from enrolling any of their state universities. Just to be sure this is not the issue- even if the application asks for a social security number, she can simply skip the question (if that is not possible in the online version of the application, she should request a paper application).
I would highly highly recommend getting in touch with a local immigrant youth organization in your state, as it is always best to talk to someone in person who has been through the process. If you share the city/state where you are located, I would be happy to put you in touch with someone close by involved in the immigrant youth movement.
Basically, there are three options for your student:
1) She can apply to a private university with need blind admissions, that accepts undocumented students in their regular pool of applicants (as opposed to treating them as international students). Over the past few years, due to the advocacy of immigrant youth, more and more private universities are open to undocumented youth and cover the cost of education through private financial aid. She has to do research to figure out the policies of private schools she is considering, with respect to undocumented students.
2) Depending on the state she lives in, she may be able to apply for a 4 year public university in her state school system and qualify for in-state tuition and/or state financial aid. Sixteen states—California, Colorado, Connecticut, Florida, Illinois, Kansas, Maryland, Minnesota, Nebraska, New Jersey, New Mexico, New York, Oregon, Texas, Utah and Washington—have laws that allow in-state tuition benefits for undocumented immigrant students. In addition, five states—California, New Mexico, Minnesota, Texas and Washington—currently offer state financial assistance.
3) She can get her associates at her local county college, and then apply for transfer at a 4 year school to lower the cost.
A few other points:
Does your student qualify for, and currently have DACA? Some schools have policies that specifically benefit undocumented youth with deferred action status. Here is an educators guide to DACA.
There are also a number of scholarships open to undocumented youth (some are specifically targeted to them). You can find an incomplete list here.
United We Dream, a national network for undocumented youth, has more information and resources through the Dreamers Empowerment through Education Program (DEEP).
Lastly, thank you for the important work you do. Please let her know that there are thousands of students out there like her, many of whom would be happy to share their experiences. And not to lose hope!