Answer:
illegals can get married anywhere to whoever they want, is no big deal.
As the ability to get married is a State-level jurisdiction, and immigration status is a Federal-level jurisdiction, the two are NOT related.
I am not aware of any state that requires a person to be a legal immigrant to be married, though they almost certainly will have to produce some sort of identification document to prove who they are (NOT that they are here legally). For instance, a person who has overstayed their visa is technically an illegal immigrant, yet would still have a passport (from their home country) which could easily be used to fulfill the identification requirement necessary to get a marriage license.
However, do note that marriage does not automatically convey any change in immigration status. That is an ENTIRELY different question.