Seconded! Filemaker is distinctly inferior to MS Access, and still proprietary/non-free. (And not especially inexpensive either.)
I wish I had a better approach to offer. Of the two, Access is a MUCH better option. Excel is also better.
I was actually in this same boat about a year ago. US-based (so somewhat less complicated) non-profit needed a system for tracking paid memberships. I looked at various off-shelf CRM-type packages, church congregation management software, and a few related fields. The complexity level of all of them was orders of magnitude beyond what they needed or what they’d be capable of learning; and yet they still managed to fall short of a few of their more unique needs. Implementing any of them would have required some degree of customization (IE bespoke coding) which would have complicated upgrades, reduced others’ ability to maintain them, etc., all while leaving them with a complicated beast they’d never really understand.
I ended up writing something in ${PROGRAMMING_LANGUAGE_OF_CHOICE} (doesn't make any difference what language you choose - nobody on staff for the org is a coder) and hosting it on OpenShift. It’s still in use, and it’s needed relatively little maintenance, but it’s definitely the kind of creation you’re going to be paying child support on for a long time. I’m always on the look out for something simple off-shelf that will do what they need for membership tracking and not be “mine,” but the available software isn’t materially easier to maintain than what I built and being orders of magnitude more complicated to use is a deal breaker.
As far as lessons learned... I'm a Java coder by day (go ahead, get your free shots in... I can take it...), and decided to do it in that as a convenience to me. We use Wicket framework at work, and I used that with Tomcat and MySQuirreL as DB. The experience of writing it was pretty good all things considered. It's well-architected (IMHO...), clearly written, little to no design debt. I took the time to clean up after myself since there wasn't really a deadline, so it's really just what they need but reasonably easy to extend if need be.
The problem with it is free or even reasonably priced Java hosting is a bit hard to come by. They have a (small) tech budget, so free wasn't an absolute requirement, but cheaper was definitely better. I tried AWS initially, but the tiny instance was too short on RAM to run the thing effectively, and it was way too slow and not especially cheap. The bigger instance sizes blew the budget completely. I ended up on OpenShift (which ironically is itself on AWS, but they pay the bills, not us...), and that's a little bit better performance-wise. It's still not super fast, but it's a back-end only system. It runs well enough. I'm still concerned about relying on a free/beta service that could go away; but I'll burn that bridge when I come to it.
In hindsight, I should have done it in PHP so they could run it on their Dreamhost site (also by no means fast, but at least paid for). It was one of those decisions where the value of the free time I was donating gave me a certain amount of leeway to take the path of least resistance (for me the programmer) at the expense of more difficulty hosting it. I don't think that was the right decision, and I'll probably end up redoing it in PHP at some point.
To summarize:
I'd say if you can possibly distill their needs to something simple that will fit in a spreadsheet, S/O/L Office (I like that...) is likely to provide the longest useful life for them and the least amount of support for you. If their needs really and truly can't fit into a spreadsheet, honestly they're getting to the point where they need to scratch up an IT budget or simplify their needs to meet the reality of what they can afford.
If you MUST develop something bespoke, the worst thing you could do is choose ${FRAMEWORK_OF_THE_WEEK} or any environment that needs more than a minimal bog standard LAMP shared hosting solution. Anything that requires you to install a machine-level framework, alter the system-wide Apache configuration, etc. is going to greatly increase cost and decrease flexibility. It's against your organization's best interests to do that. The lowest common denominator for cheap hosting is LAMP, so stick with that.
If you do write something by hand, consider you're adopting a puppy. You're going to be caring for this thing for years. If you're not prepared to do that, it's irresponsible and unprofessional for you to push them in that direction.
In my case, the org is a local group with a mission I care strongly about, so short of significant unforeseen life events, I'm in it for the long haul.