Is there a way to stream MP3s stored on Amazon S3 via a Flash widget embedded in a website, or some other method?
If you're referring to "true streaming" (not progressive downloads), you cannot do it with just S3.
You need to store the file on S3, use MediaConvert to convert your mp3 into a HLS stream (I recommend HLS compared to the others), which breaks down your mp3 into a .m3u8 and .ts files, then use a player like VideoJS, and then you can play the audio as a true stream.
Or you can also use a s3 audio streaming player like https://s3mediavault.com/demo/s3-streaming-audio-player/