If an illegal immigrant doesn't have insurance and they hit your car, you have to rely on your insurance. If you have uninsured motorist insurance, you will likely be covered.

However, if you don't, your only other avenue is to ask them to give you money for the damages, or to sue this illegal immigrant. It's highly unlikely this person has money to pay a judgement against them, and since they're illegal, they probably don't have official wages to garnish. Aside from that, you're unlikely to see an illegal immigrant appear in a court for a civil lawsuit, because they are here illegally and are not likely to voluntarily participate in a legal proceeding.

Realistically, you're probably going to have to eat this cost.