How to go about it is usually where it starts to fall apart.
The idea has come up many times, and the conversation ends there until the next time someone has the idea.
If you aren't already intimately familiar with the lore, you end up just quoting the official websites and easily-found flavortext ... and then it stagnates. If you are, your evidence and connections are usually exhibited by quoting small bits of well-buried dialogue and citing their origin to build a coherent whole. Personal lore collections are often a mess that only the one who compiled it can make sense of. "How do I build a good lore database?" has few answers that aren't equivalent in task to starting yet another Wiki / Forum / Website and your hopes of rallying the community are more likely to yet further divide it (and XIV's community is pretty fractured as-is), and, again, things start to stagnate once the novelty fades.
For me, even when I experimented with building a lore database on Gamer Escape's wiki, I ended up essentially making another wiki on a wiki. For me, it always comes back to, "Well, we already built the foundation to account for mechanics; I should just bug Gahoo to tweak it to handle accompanying lore." Story resources come from all over the game, collected manually by a community. A traditional wiki seems like the easiest way to accomplish anything, and the guy has never turned me down when I've found a way to improve how GE handles lore (except, after six years, I get to annoy him as a staff member, now).
The challenge is finding people that want to help build it instead of just showing up and being mad it's not already finished... Since ARR came online, reading of the wiki is as active as ever, but rebuilding the community that evaporated after 1.0 tanked and SE made these forums has been problematic. For what it's worth, if people want to work and build, I know they'd be happy to accommodate pretty much whatever people think it needs to make it work. I ended up with them by virtue of the fact that they just never gave up on the game, though, so I admit my two cents are biased by my respect for that, lol.