Since you tagged this Android as well- not on that OS. The keyboard is its own app and handles the voice input itself. There is no way to access the files of another app.

If you want to do this in a native app, put up your own microphone button and use the speech to text service, which will return an array of possible inputs with probabilities. In a browser, you're just out of luck as there is no access to the service.

All of this is kind of a moot point anyway for a few reasons

• Very few people use speech input. My last data on numbers is old, but it was unpopular enough when I worked at a keyboard company we had an option to remove the key.
• Uploading those files would be a huge privacy concern. Look at the firestorm a year or so ago when it was found Google/Amazon did this same thing for the same reason. This was a bigger deal in their case as it was background processing, but users would likely still not be happy.
• Unless you're spending a few million on researchers, you're not going to be doing it better than the existing solution. That kind of software is not easy to write, its not a totally solved problem even by Google and Nuance (owners of Dragon which powers Siri, or at least did) who have huge teams. Why do you think you'll do better? Unless you plan on listening to them manually. In which case the next point is even bigger.
• Ok, so you upload the file and you do find a better solution- what are you going to do about it? Somehow change the text the user typed in 20 minutes ago? How are you going to do this and have a UX flow that makes sense?