This sample page makes an AJAX call to https://www.xwordinfo.com/JSON/Data.aspx to get the data for the most recent NYT puzzle in JSON format. It should work on any modern browser. View Source to see the code. Click a button to make the call.
An implementation of this code on your site would require additional work on your side since cross-domain AJAX calls are prohibited for security reasons. In particular, this might mean writing proxy code on your server.
Or see Sample 2 for a completely self-contained jQuery version that can run from any domain with zero server code.