POST data to another website, keep user on my site

I wasn't exactly sure how to approach my situation, but I'm looking to POST form data to another website while the user stays on mine. The site I'm working on is using PHP, and have tried to play around with cURL but not getting too far so looking for some advice.

As an example, I want a user on my site to be able to order whatever they want from say (example) Papa John's mobile page https://order.papajohns.com/mobile/index.wap?site=wap then checkout on my site. The form is on my site, user info is added then a submit button is clickedto go to the next step. If I use the link to the form page for Papa John's it takes them offsite, if I try cURL it keeps the user on my site but I'm not getting the response (store locations) I want.

I've played with iframes and it works nice, but not sure I can actually store the data or fetch the totals with iframes.

So what I'm asking, is

1. Is this even possible?
2. Am I on the right mindset by thinking cURL will be able to do this?
3. Is this something that should be done in Javascript or Ajax instead?

Any help would be greatly appreciated! Thanks

Most sites don't want you to do this, so will make it difficult. I run ecommerce sites using my own code and I know I wouldn't be happy with it.

You should discuss it with the sites in question and see if they have, or can develop some kind of API

That's not an option in my case and didn't answer any of my questions, but thanks for the response.