Letter of ineligibility from the Social Security Administration.

I'm not quite sure what you are asking. IF you are asking can you get an ineligibility letter from Social Security for a disability claim you were not eligible for, you should be able to call them and have them send you copy of your last denial letter. If it is a letter of ineligibility for another benefits from Social Security, I would think you could still call them and ask them to send you a copy of that letter. IF you are wanting a letter of ineligibility for a Social Security number, I am unsure whether they would do that if your immigration case is not granted yet. It seems it would be obvious you are ineligible if your immigration case is not granted so they would probably not have a letter to send you. You may still call them and see if there is something they can send you saying you do not have a Social Security number. If you can clarify your question, someone may be able to help you more thoroughly.

You question is not clear. why do you need this letter?

Yes, it looks like you tried to get money from social security without having the proper legal status.