Short answer: no
Long answer: What audio format is the flash application producing? is it live sound or recorded?
What you would need is for the flash application to record the sound in a compatible format for the browser you are targeting and produce a blob out of it. From the blob you can create an object url in JS and feed it to the web audio API. Problem is to produce the blob from the flash app and make it available to the browser in a compatible format and I am not aware of any flash API that could do that.
A pushy workaround would be to send the recorded audio to a server to be transcoded and then hosted so that the web audio API can ingest this audio file.
Here is a good article to start with the web audio API
Best option would be to go 100% flash free but it does not seem to fit your project requirements.