How to proceed getting a Smart ID card reader for old and new MacBooks Air?
A CAC (Common Access Card) or Smartcard reader is just a USB device that reads an authentication device in much the same way a USB SD or CompactFlash card reader reads those media.  The age and/or model of your Mac is irrelevant.

What’s important is the OS.  Most card readers are compatible with macOS, But Catalina introduces some changes; specifically the change from the legacy TokenD to the new TokenCryptoKit framework.  More information can be found in the Apple Support document Prepare for smart card changes in macOS Catalina.  Basically, you may need to configure Catalina to use the older framework if the software you’re using hasn’t updated.  Your IT support staff will have to address this for you as they will be the ones who know if this is the case.

As far as what Smartcard reader to get, there are lots on the market, but I prefer to purchase from known brands; the IOGear GSR202 looks to be a good, reliable product.