I assume that you are talking about writing a script on your server that will be interacting with someone else's script on a different server (for example, maybe Amazon.com or something).
Yes, this is possible but it will take some investigation on your part. You will need to go to the first page in a web browser and do a View Source. Cut and paste this to a text editor and remove anything that isn't Javascript or Form code. All the Javascript does on the first page is manipulate Form variables. Study the Javascript to see exactly how it manipulates the Form variables. A good testing technique is to change the Form's Action field to a script on your own server (instead of theirs) and then print the contents of the $_POST variable. This way, you can see precisely what that page passes to the second PHP page.
Once you know exactly what gets passed to the second page, you can write your own script that uses fopen() to contact their 2nd PHP page (not the first page) and you pass in what that page expects to receive. When you use fgets() to read the results, you will have the web page that they return.
I know this sounds complicated - it's easier than it sounds - but it will take some work.
Some companies try to prevent you from doing what you are trying to do. (They only want you to post from their form, not from your own script). They will make Page 1 post to Page 2 which redirects to Page 3 which checks to see if your session variable shows that you were actually on Page 1 a few moments ago. If not, then the return an error.
By the way, so that you understand, there is no such thing as a PHP Post - just a "post". The second page doesn't care what language the first page was written in. The first page could be HTML, PHP, Java Server Pages, or even Fortran. It constructs an HTML Form that "posts" the fields to the next page. The next page receives the post (in a protocol called CGI). A post is a post is a post. It's a stream of data from the first page to the second page. When it arrives, it's just a string and it doesn't matter what language was used to build the form that asked for the data. (Which is why Javascript can be on that first page without the 2nd page caring how the fields got set).
