Does Anyone Know How To Find Out What Payment Processor A website Is Using If Not Advertised

Short answer is no, you can't. But you can do a few different things to get closer to the information.

You can try enabling developer tools in your browser and monitor the HTTP requests to see which URLs are being connected. But that might not tell you much, if they are using a payment gateway with many different providers.

Another is to make a payment and look at how it appears on your bank statement. Does it say the URL/company you bought from, or something else?