Finally, someone else is on the same issue. We're all brothers in arms here. For some reason, almost as if by design, Shopify support/admins on here will never give you a straight answer.
I'm looking for the same thing too. Last I've read on here, you have to "What you can do is from your front end, delegate to your backend to make the request, and return the response data from that request to your front end." Unfortunately, that's very vague. They're probably just going to tell you to hire an expert, and not answer the issue publicly.
Yeah, there seem to be some "ways" floating about such as using JSONP but they don't seem like a solution but more a "hack" to get it to work for cross-browser requests.
Apps out there have done this look at Privy, Ultimate Special Offers etc they all have on-page modals and such which require talking to their app from another domain. It's just not well documented seen as it's meant to be a big thing the Script Tag.