Of course, sorry but — rephrasing slightly for clarity — has it been their bailiwick since they were created? Like, is illegal voting an immigration law violation, or is this just Trump projecting a new responsibility onto his independently funded army?
Adjudicating whether an individual is eligible to vote is not and has never been the domain of any federal agency. It is categorically none of their business. The only federal intervention in elections that the Constitution permits is for the executive branch, traditionally through the Department of Justice, to ensure that states are not improperly preventing voters from voting, via the 15th, 19th, 24th, and 26th Amendments.