How can I sponsor my sick sister if I am a US Permanent Resident?

There is no immigrant visa category that allows a permanent resident to sponsor a sibling. Assuming that you are able to make provision for her medical treatment in the US which she is incapable of receiving in Mexico because of the more advanced state of medicine in the US, you may be able to have her apply for a visitor visa to come for medical treatment. Due to the limitations of the Lawyers.com Forums, Alan Lee & Arthur Lee, Esqs.’ (the "Firm") participation in responding to questions posted herein does not constitute legal advice, nor legal representation of the person or entity posting a question. No Attorney/Client relationship is or shall be construed to be created hereby. The information provided herein by the Firm is general, and requires that the poster obtain specific legal advice from an attorney. The poster shall not rely upon the information provided herein as legal advice nor as the basis for making any decisions of legal consequence.