Immediate relatives of U.S. citizens (children, spouse, or parents) have an application opportunity that many other people who are eligible for a green card (lawful permanent residence) do not. As long as they entered the United States in valid visa status (and didn't misuse that visa, secretly intending to apply for a U.S. green card after arrival), they may use "adjustment of status" as the procedure by which to apply for their green card. They can submit their green card to, and attend their approval interview at, an office of U.S. Citizenship and Immigration Services (USCIS).
Most would-be immigrants to the U.S. must, even if they are already living in the U.S., complete the last portion of their green card application through a U.S. consulate in their home country. There, they attend an interview and are given an "immigrant visa" to the U.S., which basically turns into lawful permanent residence upon arrival.
The ability to instead adjust status in the U.S. means, for some immigrants who are living in the U.S., the difference between being able to successfully apply for a green card or not. The reason is that they have accrued 180 days or more of “unlawful presence” in the U.S., as described in Consequences of Unlawful Presence in the U.S.: Three- and Ten-Year Time Bars.
Their departure from the U.S. could result in a time bar upon return, for three or ten years (depending on the length of the unlawful stay). Although there's a waiver available based on extreme hardship to qualifying family members, it's not easy to qualify for this waiver.
As a result, many applicants for permanent residence who have lived in the U.S. without permission have an important reason to try to adjust status. And even after an overstay of their visa or permitted time in the U.S., applicants in the "immediate relative" category can do just that, provided they can prove their lawful entry.
As you will see on the instructions for the green card application (Form I-485 being the primary form), you are expected to include proof of "Inspection and Admission or Inspection and Parole."
For most people, this means a copy of your stamped passport pages and U.S. entry visa. It also includes what's known as a Form I-94 Arrival-Departure Record, which was at one time a small card physically placed in one's passport by border officials, though it's now simply information entered into a database.
The I-485 instructions warn that, "If you cannot produce this primary evidence, and DHS has no record of the admission or parole, USCIS will presume that you came into the United States without admission or parole."
You wouldn't be the first to lack evidence of your lawful U.S. entry. For example, some children never had their own visa, but were named on a parent's B-2 tourist visa, which document was lost after the passage of time and possibly the death of the parent. Or you might have had your own documents, but had these lost or stolen. Or a border guard might have spoken to you and then waved you through, without asking to see documents.
You are not necessarily out of luck in any of these situations, but it will take some creativity to prove your legal entry.
Your first step should be to do a Freedom of Information Act (FOIA) request, which essentially asks the U.S. government for a copy of your immigration file. (Go to How to Get a Copy of Your Immigration File (FOIA Requests).) This might yield copies of your visa and I-94. If it doesn’t, your next step depends on when you entered the United States.
If your entry was after April 2013, then you probably didn’t receive a paper I-94, but were entered into a computer database. Getting a copy of your I-94 (which should be sufficient to prove your lawful entry) should be easy: Go to the Customs and Border protection website, at the page called Arrival/Departure Forms: I-94 and I-94W, and follow the links from there.
If you entered the U.S. before April of 2013, then you likely received a paper I-94. You will need to ask for a replacement, by filling out U.S. Citizenship and Immigration Services Form I-102 and paying a fee. Submit this request by mail, following the instructions on the USCIS website.
If your passport and similar documents were stolen, it would also be a good idea to include a copy of any police report that you filed, to help explain why you don’t have the originals.
If none of the above works, think about alternative forms of proof. If, for example, you arrived legally as a young person, along with your parents, you might be able to find evidence of your parents’ and your physical manner of entry into the United States. It’s probably a long shot, but if you arrived by plane, might you by chance have a copy of the tickets your family traveled on, or other receipts from your travel to the United States? (It’s nearly impossible for an undocumented person to arrive in the U.S. by air.) Do you know who picked your family up at the airport? Would that person be willing to sign an affidavit swearing that you arrived by plane?
You might also look into whether any lawyers were involved in your or your parents’ obtaining the U.S. entry visa. There’s a chance the lawyer would still have your records on file.
Finally (and as a last resort, because personal statements are considered less convincing than other forms of evidence), if anyone else was close enough to you or your family to know details of your mode of entry to the United States, that person could write a sworn declaration or affidavit explaining this. You can prepare your own statement as well, including plenty of detail so as to be convincing.
Although it is important for you to think carefully about the possible types of evidence you might show to prove your lawful U.S. entry, it would also be worth hiring an immigration lawyer to help you with this application. Your success with the adjustment of status application depends on pulling all the evidence together in a convincing fashion.