An understandable concern, but my parents frequent SouthWest more than I do the SDMB, which is to say, a lot. So yeah, I’m confident they’re spelling it correctly.
I also had them try to access the site from the Wii’s browser, and had the same problem, which should rule out it being a computer/browser problem. They’re going to have my sister bring over her laptop later and connect it directly into the modem to see if that works.
Reset the router (just turn it off for 30 seconds then back on…may take a minute or two to sort itself back out). Have your computers and Wii off while this happens. Once the router is back up turn them on and test.
If the Wii is doing it too then the problem almost has to be at the router (which may well include a firewall) or the ISP (and of course the ISP will do its level best to tell you all is well with them and that goes double for Comcast).
Another thing to check is the MTU setting. If the MTU setting of the computer/router is too high, it will cause some websites to be unreachable. I have had the same issue (although it was more than just a single website that was unreachable).
Here’s a link with a description of how to fix it. I fixed mine by changing the MTU setting on my router instead of the computer, which was easier.
Good luck.