I am a US permanent resident. How can I return to the USA after losing my Green Card in Canada? I only have my social security card on me. What do I need to cross the border?
You need your permanent resident card to return to the USA unlike a citizen who can just show up at the border with nothing and will be allowed in after some checks.
You must file Form I-131A in person at a USCIS international field office or with a Consular Section at a U.S. Embassy or Consulate. Before appearing in person, you must submit the filing fee through the online payment system on our website (see the Filing Fee section below). When you appear in person to file Form I-131A, you must bring evidence that you paid the fee
Note that it’s not cheap, $575.
What documents, identification, and paperwork does a Lawful Legal Permanent Resident (LPR) Green Card holder need to travel internationally?
Lawful Permanent Residents (LPR) of the U.S. must present a Permanent Resident Card ("Green Card", Form I-551), a Reentry Permit (if gone for more than 1 year), or a Returning Resident Visa (if gone for 2 years or more) to reenter the United States.
The US Embassy in Mexico (which also frequently sees cases like this) has a web page about this situation. They say that you can go to the land border and present your case to CBP there. If you have other documents which identify you (e.g. passport) you should bring them so that CBP can be sure of who you are.
In relevant part:
Please see below for your four options for returning to the U.S., presented in order of processing time, from least to most time required:
Have your airline call the Regional Carrier Liaison Group in Miami and seek permission to board without an entry document.
Cross a land border and present your case to CBP there. You will be able to apply for admission to the U.S. with CBP without your LPR card.