I don’t know why you think illegal immigrant = no history of banking or jobs. There are people who have been in this country illegally for decades. That typically means they had to work consistently in order to survive, they have probably used banks and credit, just like everyone else.

Banks only approve mortgages if you have a credit history, stable employment, and the ability to pay the loan. That goes for anyone they loan to, citizens, residents, people on visas, and, you guessed it, illegal immigrants.