They won't use a Chase account. They'll get paid cash or a physical check they can take to a check cashing place.
IIRC banks need a ITIN or SSN + passport or ID of the UBO
Undocumented immigrants are far more likely to go the... undocumented route.
In terms of immigration, undocumented typically means you don't currently have a valid visa or residency permit. In that case no, they would still be undocumented, since the ITIN doesn't grant them either.
(Employers are supposed to verify ID too, but they're not scrutinized as heavily as a major bank like Chase.)