There is no way to do this without a fairly extensive modification. WordPress uses native emoji, and there’s a script to convert those in the browser to the img versions if the native versions aren’t up to par.
But since that script was made, native support has come a long way, and so it no longer has a lot of things to do.
The reason you can’t do this easily is because the replacement script works on the browser side. WordPress has no idea which set of emojis are being used, nor does it care. It’s just storing native unicode text.
