What you're trying to do doesn't make any sense. When you access a website, you do it via HTTP over TCP/IP. In between there are routing devices that only forward IP packets to their destination but (simply put) don't care about your MAC address.
Well… they don't even have it, because your MAC address is only relevant in your local area network, which ends at your home gateway: the router/modem your ISP provided. After that, you will be "identified" via your public facing IP address only, and only the MAC address of your router is visible to the ISP network.
Your only alternative here is Tor, which actually can change the IP address you use to access websites, by routing your requests through different hosts.