Offering loans to people that couldn't otherwise get one increases the pool of buyers, driving demand and increasing prices. I don't raise that as a reason not to have the program, but you are missing second order effects if you consider there to be no cost to the broader public.

More importantly, how exactly is an undocumented immigrant paying taxes? Do they somehow have a tax ID or SSN without documentation? Or are you just referring to things like sales and fuel taxes?