Can I use FOIA to source lists?

Sure! It's legal. Is it wise? Um....let's skip that point for a moment. Let's start with, "it's legal!"

What does "FOIA" mean in this context? Wikipedia to the rescue:
 The Freedom of Information Act (FOIA), 5 U.S.C. § 552, is a federal freedom of information law that requires the full or partial disclosure of previously unreleased information and documents controlled by the United States government upon request. 
Apart from the U.S. federal government's Freedom of Information Act, the U.S. states have their own varying freedom of information laws.
Querying a government agency to get information back is almost always referred to as "obtaining this information via FOIA" even if that's not exactly the correct governing law in some circumstances.