Is there a Web browser app for the Apple TV? (i.e. Safari, Chrome, or other)

You would be using Safari or other browser on your IOS device (iPhone/iPad) or Mac and then mirroring it's display to the Apple TV. However, if you are just looking to stream audio from radio broadcasts then you would be better off just using basic airplay (which sends the media directly instead of mirroring the screen)

Use AirPlay or Screen Mirroring on your iPhone, iPad, or iPod touch - Apple Support

Use AirPlay to display video from your Mac on an HDTV - Apple Support

No. You do have the option of mirroring from another device (IOS or Mac) if you are really in need of browsing.