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Where the rumor began...
Snopes references an interview on The Dinner Program were Kimbal mentions the brothers were "illegal immigrants"

The meme likely originated with an appearance Musk made on The Dinner Program. While Musk's brother Kimbal once joked that they had been "illegal immigrants," Elon said that this was a "gray area" (and indeed, American immigration policy in all its complexity contains a good number of "gray areas") and that he considers himself a legal immigrant:
KIMBAL: When they did fund us, they realized that we were illegal immigrants.
ELON: Well...
KIMBAL: Yes we were!
ELON: It was a grey area.
Musk also stated that he was a "legal immigrant" during an episode of PBS' Think Tank in 2007:
WATTENBERG: You have been quoted as saying — Elon you came here from South Africa at age 17, is that right.
MUSK: Yes, I did.
WATTENBERG: You were an immigrant — a legal immigrant ...
MUSK: No, no — yes, a legal one. [Laughter]
WATTENBERG: I got to think a lot of illegals add a lot of value to the United States. You have been quoted as saying that you are nauseatingly pro American.
MUSK: Yes, that's true.
While the Tesla Motors co-founder spent more than ten years in the United States prior to becoming a citizen in 2002, it appears that he did so legally through various visa programs.
https://www.snopes.com/fact-check/elon-musk-illegal-immigrant/

Fact Check: Are Attorneys Investigating Elon Musk Over U.S. Citizenship?
Published Dec 14, 2022 
 "Just so everyone knows the facts, while I did make a FOIA request for Elon Musk's immigration application he has to consent to the request before I can get the docs.
https://www.newsweek.com/elon-musk-being-investigated-over-his-us-citizenship-fact-check-1767025
https://x.com/DeanObeidallah/status/1587058771079094273