No, a Lawful Permanent Resident (green card holder) cannot sponsor a K-1 visa. The K-1 visa process requires the petitioner to be a U.S. citizen. The U.S. citizen must file a petition on behalf of their foreign fiancé(e) and meet certain eligibility criteria, including proving their intent to marry within 90 days of the fiancé(e)'s arrival in the United States and the ability to financially support their fiancé(e).