Hi Mark
You wrote:
> The school wants to use their offline LMS in their school
The off-line LMS I can follow. The rest you need to explain this more carefully.
> and after the school period (as the students/teacher went home) they want to access their LMS online
Here "they" here means the administrators of the school, right?
And they have Internet access, all the time?
And the second LMS, the on-line LMS, is somewhere in the Internet.
> where the assignments and activities they have in their school was already synchronized in the online moodle server.
So they assume, by the time they access the on-line LMS, it was synchronized with the off-line LMS. How, if the off-line LMS is off-line?
I don't know, perhaps you mean that the school connects its LAN temporarily to Internet via a dial-up connection and during that time the off-line LMS pushes its data to the on-line LMS?
The background is that Moodle architecture does now foresee two Moodles melting in to one. But if you mean, one is the snap-shot of the other, taken when the other is inactive, that can be done. See
https://docs.moodle.org/en/Moodle_migration. Taking the snap-shot can be seen as a Moodle to Moodle migration. If that is, what you are looking for.
There were many discussions of more sophisticated approaches. Examples "How to replicate in moodle?"
https://moodle.org/mod/forum/discuss.php?d=373346, "Moodle Installation in Rural Area"
https://moodle.org/mod/forum/discuss.php?d=353602. You need to search the forums here for more.