@khurtwilliams There are ways you could work around that. Temporarily connect your device to another network while signing in (like your smartphone's hotspot). Use a VPN. Manually copy the sign-in link from another device (with access to your email). Forward the sign-in mail to an internal inbox that you can access from the locked down network.
But, of course, the best solution is to leave that network and never ever return to it. It sounds like a horrible situation.