You really need to discuss this with the motel's IT. It's not the computer's problem unless you have turned on the firewall or have proxies installed.
I've had similar problems when on the road. I just use a browser access to the webmail access. Usually that's less time consuming than trying to "fix" it at the motel.
Why does it happen and what would I discuss with the motel's IT? Seldom do I find that they even have one. I really don't want to use my browser as I have too many accounts in too many places, and I use Apple Mail to track and record all my replies, etc.
Sure seems there could be another fix?