[Mark McLeod]

The basic network aspects are relatively easy.
The robot traffic cannot pass through a gateway, i.e., it won't go out on the Internet, but will get routed through any local area network combination of switches, access points, routers.

The problem you'll likely encounter with a school network is whatever network security measures the administration enforces.
The hardest, possibly insurmountable, part is that the scattered school access points would have to recognize, accept, and connect automatically to the robot bridge.
That means giving the bridge the SSID that the school is actively looking for, and will automatically allow connection to, implementing whatever security IT may require. You'll need to clear this with the IT department and get their active participation.

Is the robot bridge or Driver Station laptop allowed to connect to the school network?
Does connecting to the school network require a login process, like a student id?

P.S.
I know one team had to mount a laptop on their robot to use as a pass-through, so they could logon to the school network.