Again, it would really help to know the exact model of your Mac so we can provide you with options available specifically for your specific Mac.
How did you attempt to access Internet Recovery Mode? If you want to install an older version of macOS which originally shipped with the Mac (or the oldest online installer available for that Mac), then you must access Internet Recovery Mode using Command + Option + Shift + R. In my own personal experience and from other forum posts, Internet Recovery Mode is sometimes broken for some systems, but in those cases the only online macOS installer accessible is the oldest one...it doesn't seem to err in the other direction. Using just the general Recovery Mode (Command + R) option, can result in any of several different online installers (usually the current/last OS installed, or once again just the oldest one...I'm not certain what Apple uses to determine which one for general recovery mode). Make sure to use a wired USB keyboard to access Internet Recovery Mode or Recovery Mode (in fact when using any special startup keys).
If Mavericks is not available under "Previous Purchases" in the App Store, then you will have to choose one of the other existing versions of macOS which are available for download....certainly one of them should work for you since a driver is rarely restricted to just a single version of OS.
The easiest option would be to use macOS 10.10+ since you can create a bootable macOS USB installer (even using this laptop by first installing High Sierra if necessary if that is all you can access). I'm doubtful it will be possible (or easy) to make a bootable macOS 10.7 or 10.8 USB installer. Otherwise if you are unable to make a bootable macOS 10.7/10.8 USB installer, then you will need to install the oldest online installer so you can then download & extract the 10.8 installer. The former is my personal preference and what I would try first....without any knowledge of the driver or its specs, I personally would go for 10.10 since I don't think much was changed with it (I think 10.11 changed a fair amount, but possibly may still be compatible with drivers..in my mind it would be worth testing).
If the laptop is compatible with OSX 10.6, then using an OSX 10.6 DVD (depends on the exact model of Mac which I don't know) is an option which you could then use to upgrade to 10.7 or 10.8.
The only other option I can think of if macOS 10.9 Mavericks is the only OS the driver will work....then get access to a Mac which originally shipped with macOS 10.9 (probably some 2013 or 2014 models) so you could use Internet Recovery Mode to install it. You could can clone that installation using Carbon Copy Cloner (CCC) to another drive for the Mac you want 10.9 on.
https://eshop.macsales.com/guides/Mac_OS_X_Compatibility
I agree it is unfortunate that under some circumstances Apple makes it very difficult to access & create macOS USB installers, but this is all we have at this time and it is unlikely to change. It would be great if Apple provide access to macOS installers from outside the App Store which could easily be made into bootable USB installer using any device or OS instead of all the restrictions, but this seems highly unlikely to ever happen.
You may also want to consider creating another thread asking for assistance in getting the older hardware to work with newer versions of macOS just in case someone may have some ideas. Of course it may be impossible, but it never hurts to ask.