Sure, but you will have to get the details for the requirements on the server end of things. Most server-side processing programs will expect the data format to be URL-encoded and use the POST method, which is most often what HTML forms use. In Acrobat, you'd set up the submit action to submit the data as "HTML", as opposed to FDF or XFDF. A form can be set up to submit as XML as well if the server is expecting that format.
Also, information is often passed to the server using hidden fields, so the details will tell you what values to use, as well as the field names to use for the form.
It really depends on who the payment provider is, and what you mean by "process" - are you collecting card details and submitting them to a website, or
is it an interactive process where the server sends back information? There are also restrictions imposed by most banks as to exactly how and where the data is collected (for example some credit cards will run interactive verification using an iframe in the host website, and that can't be done via a PDF).
Before investing any time in the work, I'd suggest talking to your provider and checking what's allowed, and what's technically possible. Those which do allow remote submission will often have an API layer that's easier to talk to.