Is there a good application for Mac to manage subscriptions billings?
It’s not 100% clear what is the meaning of “automatize the payment process” but a very popular Mac application for billings is called… Billings.

There are a few “comparison” reviews out there and you will not have a hard time finding there. Some are more extense (and outdated) and others are short and concise. There are also some round-ups about invoicing apps (which are usually married with billings apps).

I personally use GrandTotal ($) because I needed an Invoicing app for my little company. You can try most if not all of the applications for a few days before deciding which one is for you (or usually which one is “the closest” to what you want, because they all have stuff you don’t need and lack something you wish they had…).