This seems to be completely unrelated software from
https://github.com/janeczku/calibre-web -- as far as I can tell, it treats your calibre library as completely read-only, so it should make absolutely no difference whether calibre is running at the same time, as "calibre-web" will never try to modify the database.
That being said, if you're running server software anyway, I'd just go with the builtin server in calibre. You'll need calibre installed anyway in order to create the database, unless you're syncing your desktop library to a remote cloud instance.