They just use some other means of referencing position such as IR LEDs + camera (like a wiimote or in reverse), image capture/analysis, gun position sensing, or some combination of these things. Most use IR LEDs. Some older technologies such as the NES Power Glove used ultrasonic positioning.
A lot of the 2000's era arcade gun games such as Time Crisis 4 used DLP projectors from the get-go and were using these types of gun controllers from the start; so they are relatively easy to convert to LCD.
Classic CRT based light gun games -- while I'm sure it's possible to build some sort of device that emulates the original gun in hardware, it is probably a much easier job to simply run them in an emulator.
One saving grace to this article is that while it's true that the CRT business might be winding down, the tubes themselves do usually last far longer than the electronics and will be around for a very long time still. I have had to replace some components on my Wells Gardner CRT that I used in my scratch-built cabinet because it had gotten very dim, but after a new neck board and some new capacitors it's back to looking like new.