The plugin in default settings kick all the players when join, with "Permissions data couldn't be loaded! Try again later!" message. Please fix it.

My server's version: 1.12.2
Serverfile: craftbukkit

Regarding CraftBukkit + Offline Mode + LuckPerms, please see here:
https://github.com/lucko/LuckPerms/wiki/Setup#craftbukkit-and-offline-mode

Your other issues are addressed at the very start of the "Setup" wiki page. 🙂

Please enable debug-logins in the LP config and send us the log of the debug.

You should switch to Spigot...

Correct. This is a bug in craftbukkit, not luckperms. We're currently waiting for a fix on their side. For the time being, please switch to either spigot or paperspigot (I personally recommend paperspigot).

If the issue persists after switching, please do as brainstone says, as well as send us your latest.log via gist.github.com.

Ok, thanks, I try paperspigot.

The plugin is not work in Spigot. I write a command in chat, and the plugin says me "Running LuckPerms v.4.0.18"

Please fix the problem, and kompatible it for craftbukkit

The message you get is normal, if you don't have permissions for LuckPerms.
Just give yourself luckperms.* through console and it should work...

Also reading the wiki solves a lot of problems.

As for making it compatible with craftbukkit, we simply can't do that. We must wait for craftbukkit to implement the fix on their side, just like spigot and paperspigot have.

Anyway, you should be good to go now. Just follow the directions laid out on the Usage section of the wiki. And if you need additional help regarding the setup of luckperms after reading the wiki page, I recommend you join our support discord. Support will be faster.