American Presidency Project: Franklin D. Roosevelt
Contains audio and (when available) video of FDR's speeches and fireside chats
Online archives at the Presidential Library and Museum
A smaller number of speeches (and one video) are available online here
YouTube
A number of FDR's speeches and public appearances have popped up on YouTube, including fireside chats, and a speech before Congress. Watch out, though; some have been, shall we say, modified.