How can I receive FM radio broadcast signals on an iPhone 4? I know I can download apps such as Wunder Radio or iHeartRadio that tune in to the same signal over the internet as is being broadcast over commercial FM stations, but I want to use my iPhone to receive input from a real FM tuner.

Many times smaller stations do not stream and often other services, such as audio of TV on a Gym, some museums and traffic warning stations as well as conferences where live translations in several languages are broadcast over different FM frequencies. 

Is there a particular app for this or maybe a third-party device I can purchase?
The only iOS device that features a built-in FM tuner is the iPod nano, since the fifth generation (the tall "candy-bar" style before they shrunk to the square shape). The sixth generation (current, as of this writing) iPod nano still has the FM Tuner.

It is impossible to merely download an app on any iOS device (iPod touch / iPhone / iPad ) to accomplish this with any modern/iOS device because you have to have the physical radio that can tune FM/AM radio bands. Now, if they streamed to some network location which an app can easily access...

I would like to mention also that the Griffin and Belkin products in The Apple Store ARE NOT tuners. They are transmitters. They broadcast (short range) on FM frequencies, so you can tune in using your car stereo, or perhaps some kind of radio tuner alarm clock you may have in your house.

The iHome speaker does appear to do FM tuning, but that is likely not practical in the example you're asking about. Go with the iPod nano.