No.
If you are the producer of that program, you own the content (either by explicit contract or by default).
If you are a third party (which you are), you would need permission from the content owner (producer/radio station),
and also
the artist/interview subject him/herself. Think in terms of music copyright; there are at least two copyrights in any released recording-- the underlying song content, and the recording itself. You also have to consider what copyright law calls "derivative works."
Think in terms of photos and
graphic images; if someone snaps your picture during a news event, they don't need your permission to submit that photo to the New York Times, the local newspaper, or their own news-related blog or website. But, if that same photographer wants to use your image in an advertising campaign or on a cereal box, they'd better get your permission and poney-up some dough, right?!?!??
This (your project idea) is basically something that
can't be done without opening you up to major legal exposure, and at
minimum, a not-so-friendly cease & desist letter.
But, whatever...
GJ