Older VFD datasheets are hard to locate, and slight differences in part numbers can indicate significant differences in interface, character set, etc. The closest that I can find is the scpb datasheet, which I guess you have already seen, from here: **broken link removed**
I've no way to tell if it's identical; my comments below will refer to that datasheet in the hopes that it is. Check out the connections to your module and see if they correlate with the datasheet, that will be a start at 'maybe it's the same'.
It might be quicker to locate a replacement VFD module. I've compared the datasheet with a few other VFD's that I have on hand though and they have slight differences in the command method (using the A0 line, and character positioning stand out) while the basic English character set and most display functions (like scroll modes, BS, CR) are similar to a couple of Noritake's. This is typical of the VFD's I have experience with; they are all different enough that code, and sometime the interface, needs to be altered for each one.
So, my method (assuming the VFD still works a little and has just gone dim or lost some lines) would be:
1. Checking if your VFD seems to match the scpb connections. If it does, let's assume it matches the rest of the datasheet too. If not, try figure out it's connections to the CPU and whether pins are input or output.
2. Pull the VFD out and program an MCU to talk to it, according to the datasheet. See if you can make it work as expected. If the VFD does not match the datasheet, this part will be much trickier, as will everything else. You'll need to try to find a datasheet for a VFD that matches the connections (search Futaba, Noritake-Itron, etc), then try it's command structure, and any others too that you can find. Start by simply sending ASCII codes.
(2a - in case the VFD is really dead). In this case, either use a logic analyser or program an MCU to analyse data on the connects. See if what it shows correlates with the/some other datasheet. Work an emulator from there.
3. When you know how to control the display (and thus how the original device does), you are in a position to start to write a translator. Basically, the MCU will emulate the VFD's connections and read the incoming data from the device, translate to an LCD form, and output to LCD.
Caveats:
If the large indicator dots are used on the VFD, you'll struggle to find an LCD that has them. You might have to design your translator to cope in some other way.
If the original device reads from the VFD (if it does, it's most commonly just to check readiness for the next character, but could be cursor position or even character at current position) then that complicates matters a little. Analysing the signalling will show if this is happening. If it's just a case of checking readiness, that's easily emulated.
If the original device sends custom characters, or uses one of the VFD's alternative character sets, then again you might struggle with the LCD choice.
...
So, it all 'depends'. In the simplest case, it's easy, but it could well be more complicated, mostly because of the datasheet uncertainty. Hope that the scpb is the same!