What is the best (cheapest, most open) way to record audio via a web page. Then keep that recording as MP3 format and make it available via a unique ID / URL.
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You can't record audio using a web page, but if for some reason you need to embed a sound recording destop app (as you've said, using a plug-in of some kind) then a Java applet is probably the best/easiest way. A custom plugin/extension would be better, but harder, unless there is one already (there may be).
You should probably also reconsider using a web page for this. You haven't really given you requirements, so I can't say, but it may be much better to use a network application other than your web browser :)