This is TECHNICALLY possible*, but not the way you described in your post.
It would take a lot of work, and could face some issues with moderation but it is technically possible.
How you would do this, is you would set up a website / web server and connect it with post / get requests. So I.E. a user would click the button, then it would send a post request to the site with the input of the players name or ID or whatever.
Then ensuring that the user is logged into the site, with the correct permissions it would record their microphone for the length of time. Proceeding this, when the user clicks to stop it would send a get request. Your site would then web scrape ROBLOX with a .ROBLOSECURITY token, upload the saved microphone audio (you would have to do some checking with your web server to ensure it meets roblox’s requirements), then web scrape or use the roblox web api to get the ID, have the script WAIT until it is either accepted or denied then if it is accepted return the ID of the sound and play it. You would also probably have to have a fall back in case ROBLOX denies it.
Another way that you could do this is by instead of uploading to ROBLOX, you just play it to every user who is in the same server. This would be hard to do though as you would have to modify your server & website to accept server id’s and have functions for when the user leaves, and getting the correct permissions to send requests inbetween etc. It would actually be probably easier to do it this way, but both are extremely hard.
I hope this might be helpful if you are actually interested in learning how to do this. Anyway, bye.