Hi Michal,
As Mary has suggested, https://moodle.net/ is the official repository for sharing Moodle-format learning resources.
To broaden your search, I can suggest looking for open education resources (OER). An important thing to look for in quizzes and resources that we can re-use is that they're appropriately licensed. The most common licences are Creative Commons (CC): https://creativecommons.org/ A lot of ESL resources out there are called "open" but if we read the small-print, they aren't and we could get into trouble for copyright infringement. CC licences are a more reliable way of knowing what we can and can't do legally with the resources that we find and want to share, e.g. if we find grammar questions, convert them into Moodle Quiz module format, and re-share them.
Here's a few example places to look (I'm not vouching for any of them, just pointing out that they're there):
- http://coerll.utexas.edu/coerll/
- https://www.merlot.org/merlot/index.htm
- https://www.oercommons.org/
- https://open.bccampus.ca/
Also, you may find useful resources for grammar under "adult literacy."
P.S. I've worked for a number of organisations creating libraries of EFL and ESL learning resources and haven't yet convinced any of them of the benefits of sharing and collaborating with other organisations and adopting open licences. But that is in the private sector, where openness isn't really a thing (yet?).
As you say, there's a lot of people out there re-inventing the wheel and I think it's going to take some longer-term projects at public institutions to change that.
I have little free time but I'd like to help if I can. We could co-author some resources to put up on Moodle.net and maybe that'd encourage others to contribute?
I hope this helps!