It is obvious that a web shop including its order processing can benefit from characteristics of public cloud platforms such as elastic scalability. These assets of such a solution can definitely benefit from time-bound increase of resources during busy shopping seasons such as Christmas while at the same time do not need a lot of resources during seasons with less shopping-activity such as summer holiday season.
On the other hand in many countries personal billing and payment information are treated as very sensitive. Therefore in some scenarios you might want or have to keep them on-premises in your own data center.
Finally this situation leads to a hybrid cloud scenario where you definitely benefit from putting the assets of the front-end web shop including the order processing in the cloud while leaving your payment and billing account management services on-premises in your own data center. Based on the assumption above this simple scenario leads to two different integration steps from assets hosted in the public cloud to on-premises operated assets: