How do I let user save a file and keep editing that file in browser Javascript only?

You can use localStorage to achieve this without needing any other permission from the user.

localStorage.setItem("data", JSON.stringify(data));

If your data is just JSON then this would work, however if you have custom data types, you can take a look here.

Edit:

Since you wanted to save the file directly to the device and edit it, you can take a look at File System Access API. This article here explains it.

You can load the file first by using,

let fileHandle;
butOpenFile.addEventListener('click', async () => {
  [fileHandle] = await window.showOpenFilePicker();
  const file = await fileHandle.getFile();
  const contents = await file.text();
  textArea.value = contents;
});

Once you have the file handle you should be able to write to the file without requesting to download a new file everytime there is a change.

async function writeFile(fileHandle, contents) {
  // Create a FileSystemWritableFileStream to write to.
  const writable = await fileHandle.createWritable();
  // Write the contents of the file to the stream.
  await writable.write(contents);
  // Close the file and write the contents to disk.
  await writable.close();
}

The codes are from the article I have linked above and the article explains everything much clearly. It's worth reading.