Add the overlay to OBS

You do not need a plugin. OBS loads the overlay as a browser source.

1. Choose Create my overlay and enter your Chess.com or Lichess username.
2. Choose the mode and click Start counter if you want stats for the current stream only.
3. Click Copy URL.
4. In OBS, add a source and choose Browser.
5. Paste the URL. Set the width to about 420 pixels, or about 840 if you show two sites. Set the height to about 220.
6. Turn on a transparent background, then place the source over your board.

The same URL works in other tools that support a browser source, including Streamlabs.