Creating a Web Radio
I don't think that you can do something like this with just JavaScript and HTML5. And if we could, we would most likely have to wait a long while before clients could use it in there browser. Like David said, Flash would work, the problem with that is: A) a lot of devices wouldn't support it and B) Flash is far from free. There are other ways of doing what you described but they would require other programming or at least other plugins. The only other alternative I can think of is Silverlight from Microsoft, free and not overly hard to use, it does have mic support, not sure how great it is though. Hope this helps.
There is a specification for interacting with devices, such as microphones, but it is early days and I'm not aware of any support for it.
If you want something that can interact with a mic today, look to Adobe Flash.
Pretty much any solution you choose will suffer from bad acceptance on one device or another, and web-based solution is even worse (as extensively pointed out by everybody).
If you want to cover the largest audience possible, set up a streaming server like Icecast and broadcast over MP3 and/or OGG (both is better). Pretty much every internet connected device with reasonable capabilities will be able to play streaming audio, so let the user pick the appropriate player or its favorite one.
http://web.psung.name/zeya/ - this app transcodes your music on the fly and streams it using HTML5. Maybe this will help a little bit ;)