It’s my understanding that most credit/background is done off of linking of names to addresses. The better your information (or less common the name), the better your chances to run a good report. Also, if they are here legally they will have a green card. Green cards have a number on them that is similar to a SSN that might be able to be run through some type of system. (never tried it)
Typically when I run a report on a someone who I suspect is an immigrant (legal or otherwise), their name and addresses match. Their SSN is not a match (possibly because it’s a green card number). I’ve had both criminal and evictions pop up on reports like this. That seems to be the pattern of what they look like.
Maybe have these residents write in their green card number in place of the SSN and present their green card to the manager when they drop off their application. Just to make them jump through an extra hoop to ensure they are legal. I know we probably don’t all have a great deal of faith in the FBI, but so long as they have a green card, you can at least say that from the date of the green card back, you knew their background was clean enough to be allowed into the US legally. Whatever that’s worth is a different discussion. From the green card forward, you should be able to run your background check using names/addresses.