For what it's worth, some years ago I contributed to a French open-source booking system that went by the name of 'Openflyers'. Said contributions were to both the code and (English) translations, which I felt was the least one could do as we were using the software with a flying school I was involved with at the time. I seem to recall that for an earlier iteration of such software it was quite well featured as it included maintenance data etc on the aircraft, and once they got used to it the instructors seemed to be quite happy with how it operated in terms of booking and accounting.
Unfortunately, from about 2008 or a little later I
think the software moved on to become a closed-source system, possibly shortly after it moved from v1.3.x to 2.x (it was a while ago, I recall some discussion with Christophe Laratte at the time but it's getting fairly hazy now). Please accept my apologies if I've got the history wrong, but in any event these days it seems to operate as an online service you can purchase...
Mindful of Pilot DAR's comments (and my own views) I won't provide a link to the current online site, however a quick search showed me that version 2.0 appears to be still available from Sourceforge
here, and 1.31 as ported for PHP 7 from Github
here. There may be other more recent versions, but in any event as I'm pretty sure it was produced under a GPL 2.0 license it could be that anyone keen enough to consider producing a [open-source!] booking system may find it easier to utilise this as a base and update to fit in with today's environments, rather than completely re-invent the wheel.
Bearing in mind it seemed to work ok as it was, it may also be that one could even persuade one of the extant versions to work well enough within a VM or Docker image running an earlier OS on top of current hardware and modern OS. For a cash-strapped aero club, or enthusiastic individual this may still be a usable option. If you need to put it online then with a modern proxy in front of it, along with a few other reasonable restrictions, you might be able to address the majority of any security concerns too - although please note this is just an off-hand comment, I've certainly not looked at the code or specifically considered such a thing in many years now.
UPDATE: I see I last commented on Openflyers in 2010
here, and prior to that a couple of times in 2008, so yes it was quite a while ago!
FP.