It takes some pretty specific software to ban an IP addy from a site,
In my experience as a webmaster, it is actually remarkably easy. Go into the Control Panel and there is a link that says "Ban IP". You can ban individual ips or ranges of ips (it is possible that the webmaster is blocking a range of ip's and your ip is part of that range).
Basically, what is happening is somehow your ip can't access the website. What Proxify does is go to the website through a proxy, or a different ip address. So if your ip address is banned, then Proxify will get you through on an ip address that isn't banned. Make sense?
Of course, other things besides being banned can cause this; however, we can rule most of them out.
It shouldn't be a HOSTS file issue because the same computer could access it via a different connection (just to be sure, navigate to C:\Windows\system32\drivers\etc and open up the HOSTS file in NotePad. Is there a line for the website?)
It shouldn't be a Verizon issue because I find it hard to believe that you can access EVERY single site except for that one.
So my bet is still that somehow you're still being banned (even if the webmaster doesn't know it). To follow up on Johanna's reasoning, maybe it's a problem with the host of the website rather than the website itself.