This has always been a big question to me. My local Orthodox Rabbi (he is learned in Shulchan Aruch, a known Rabbi) told me that he doesn’t see any problem with it, obviously as long as you don’t take any action on Shabbos.
However, I did send a message to Business Halacha and I quote the response, directly from HaRav Chaim Kohn Shlita:
“In my opinion, the relevant factor is not if you are paid immediately but that you permit activating your obligation to deliver. With it, you participate creating the kinyan hischayvus. Therefore, it is necessary that your website isn’t actively processing the requests on Shabbos.
KT Chaim Kohn”