Unfortunately, any requests by the browser outside your "Origin" are possible only if there is an Access-Control-Allow-Origin in the response headers with your origin. This case is all closed for security, because if it were possible to call another AJAX from any site - a lot of bad things could be done.
Learn more about cross-domain AJAX for example here.
Websockets will not help you in this case at all, since This is a separate protocol for communication between the server and the client. And moreover, lately browsers are increasingly forcing wss (encrypted web sockets), for which besides Websockets support, SSL is also needed on this supporting server. So you especially can not do without an intermediate php script.
If you are asking this question because of the heavy load on your server - you should think about alleviating this whole business with something like the same websockets. Only not third-party servers, but your own. Which will, for example, itself, with a certain frequency, update its cache on remote servers and send information to clients with a certain frequency (or after the change). In websockets, the connection is kept and a two-way dialogue between the client and the server is possible. Thus, clients will not need to constantly refer to any page, and they will simply receive a websocket message upon sending it. This can be implemented for example on Node.js or ASP.net. In PHP, I personally have not seen such things.