You wouldn't need a VPN server at home, just a VPN client.
It seems like Verizon's telling you that their only route to that site is crappy, so if you want to get to that site, you'll have to get to it from somewhere else on the Internet that doesn't go through the same crappy route that Verizon is sending your traffic through.
So you need to fire up a VPN client on your Mac, and VPN into a VPN server somewhere else. Make sure you've set things up so that the route(s) for that website's IP address(es) go through the VPN (some VPN clients let you check a box that says "Send all traffic through VPN").
If you don't currently have access to a VPN server somewhere else, there are online VPN services you can pay for.
If you like the VPN solution, you could look at getting your home gateway router to act as your VPN client, but I don't think the Verizon FiOS (actually Actiontec) MI424WR factory firmware supports that. You can double-check that yourself via your router's web admin interface. It may be that you can find aftermarket open source firmware for your router (like DD-WRT, OpenWrt, Tomato, etc.) that supports your router and adds this feature. Or you could buy a home gateway router with built-in VPN client support. There are SOHO/telecommuter-focused gateways that make a big deal of their VPN client support.