I would like to learn how to code my own system

Good for you! Writing the system is the one sure-fire way to understand it all. However, as others have already mentioned, that is a big undertaking - especially if you are learning the underlying language and the associated web protocols and technologies at the same time. That's not to say you shouldn't try but just don't expect it to be easy or quick to complete.

or possibly better yet find a similar system that I can use

This might be an option. There were a whole bunch of such systems 15 to 20 years ago but that mode of operating (what you refer to as a Membership Replicator System) seems to have fallen out of fashion since and most of the applications in use back then have stagnated. There are probably a few still going but I don't have the details, sorry.

The person who provided the platform has gone out of business

Big question then: who owns the code to the existing system and what is the licence? If (big if - check with your lawyers if in any doubt whatsoever) you have the rights to do so you could use the existing system as a starting point. Learn how it works, understand the code and then, only then, start to go about modifying it to suit your needs. This assumes that you actually have the code of course and that existing system isn't just some SaaS offering instead.

Good luck with it however you decide to approach it. We in the Monastery can help you with the Perl side of things and probably some more besides.