The laptop I'm typing this on is connected to my home network via WiFi. And I can access the whole home network, as far as authorisations permit it.
In fact, if I'm in the office, all 1000 employees are connected via WiFi to the office network that includes servers and printers. The servers and printers most likely are connected to the network via Ethernet cables, although for some printers it's quite well possible to do it wireless.

In both cases (home and work), there's only Internet if the modem (that's somewhere in the network) is connected to the ISP (via ADSL, cable, glass fiber or satellite). If that connection is broken, we only have the local (home or office) network. The cloud and external e-mail don't function.

I'd ask your IT staff to set it up to your needs. And when that's done ask them to implement a project management system that doesn't need the cloud, so always works. Then maybe you don't need Internet.

Did you consider to use a wireless 3G or 4G dongle and connect to Internet just like your smartphone does? A subscription of 20 GB of monthly data isn't very expensive. Or is the construction site so far from the civilised world that even smartphones don't have a data connection?