Of course it's a needle in a haystack of all my 3rd party apps on Mavericks/Lion.
To work with PDF I use FoxitReader and Adobe Acrobat DC, for OCR FineReader 12.1.11. Even contacting the developer not always helpful, but you should try anyway and maybe you succeed getting them to sell licences. However if you in the desperate need of some abandoned piece of software the only option you have is tracker sites, but this is scant because most of the time seeding is no more as the majority of Mac users uses new versions. You may run into the issue that the form of the licence number you own doesn't conform to the one in an earlier version of the same app.
With the exception of Transmit I was able to find and install successfully on both OSes. Among those are Rogue Ameba (Audio Hijack), Chronos (iClipboard), ExpanDrive, Cyberduck, Transmit. Thus I was able to download installers a couple of times. You also may use WaybackMachine to crawl through old snapshots of the sites in question. Some sites offer older versions, most of them assume you're an idiot if you're running something older than 2 or 3 current macOSes (as a rule of thumb, most of devs support from Yosemite onwards, many require El Capitan and newer, or Sierra and newer), however I did find that significant faction of developers both on and outside MAS have Mavericks as the minimum and some apps in fact run on Mavericks way faster than on later macOSes and you still can buy them. Unfortunately, official sites is dead-lock. I still use Mavericks among other options and love it, I also run Lion and have a boatload of apps that make me productive on both despite them being demoted as "old". I'm sure they'll be lots of things I don't know about or can't find, but I can probably help some people! So in the meantime, if there's anyone else still on Mavericks and you need compatible software for a specific purpose, please post about it here. I was originally going to write a very long post with all the great software and compatible versions I've found, and I may still do that at some point, but it's going to be REALLY long and I don't know when I'll get around to it. Modern macOS is no longer a platform I want to use, but I'd rather go back to an older version than outright switch to Windows or Linux. I've run into this a lot over the past few months because I'm trying to downgrade my life to Mavericks. I sometimes can't even find compatibility information listed anymore-the website will just say "Works on macOS™️!" with no mention of which Mac OS. Some time in the past couple years, this practice appears to have fallen out of fashion. It used to be that when an app dropped support for an old operating system, they'd say something about it on their website, and include a download link for the last compatible version.