Docker for Windows - Settings to Cisco AnyConnect allow share driver
I haven't spent enough time to get this working on my specific environment, but this looks promising: https://github.com/docker/for-win/issues/360
They suggest changing the subnet address in Docker to something the VPN doesn't use.
Edit: If that doesn't work and your employer allows you to use another VPN client, I've had success using OpenConnect in place of Cisco AnyConnect.