If students are completely undocumented, they have no SSN, and the school district is almost certain to be aware of that fact. If they were formerly documented, and were awarded an SSN, but have fallen out of that status, the school district might not be aware of the current situation. But in that latter case, the family surely knows.
In addition, there are scads of students who are legally in the US (F1 and J1 students here independently, and F, G, H, J, etc. visa dependents) who can’t file the FAFSA.