Based on the limited details you've already given, I feel confident saying that you do not want to make your own payment processor. (Or even host your instance of one.)
Furthermore, unless you've determined that it's worth it to your organization to internalize the security costs and risks, you should avoid any solution in which anyone's credit card details even pass through any computer system under your control.
The good news is that lots of people are offering services well suited for your situation. CJM's answer lists a few targeted options. I also know at least one person who uses donorbox.
I'll go a little further out on a limb and suggest against trying to find an all-in-one service like NationBuilder for your web-host, Content Management System, Customer Relations Manager, and payment processor. Some well experienced people may disagree with me about this (I know people who specifically like NationBuilder, and it does have nice capabilities), but I feel like there's a long-term advantage to having your website management, your payment processing, and you "member" management handled by stand-alone tools.
Specifically the advantage is that at some point you're going to want to overhaul one of those systems, and that will be easier if you're not worrying about breaking the entire business stack when you do it. Also, the time you save upfront with an all-in-one solution doesn't necessarily translate into long-term simplicity; internalizing a little of the inherent complexity of what you're trying to do upfront will mean you have better control in the long run.
Since this is
Software Recommendations, I'll go ahead and assume that you're working with a small, volunteer-based, non-profit that mostly just needs to communicate who they are to the public, and get donations from some people. I'd suggest starting with
- A Jekyll based website managed through Siteleaf, possibly hosted by GitHub or whoever.
- A private spreadsheet hosted in the cloud someplace for keeping track of members, donors, and mailing lists.
- A completely external donation system, take your pick.
Those are all small tools but they work, and they're easy to upgrade if you outgrow them.