Yes, computers can have MAC addresses, and they are unique. But those are physical addresses for talking over ethernet networks. IP based commmunication, like on the internet, is a horse of a different color. If you're actually being blocked, then it is indeed based on your IP address. The web server does not know, or care, about your MAC address.
First, you've narrowed it down to a single computer, but you're not narrowing it any further. There is no magic "file" that can be put on your computer to do that sort of thing.
If there was an issue on that computer, then using a different web browser would clear it up, because the issue would be confined to the browser. Browsers are solitary, they don't interact with each other. So if you had the problem in Chrome, then you would not have it in Firefox.
Alternatively, perhaps the issue is not in the browser, nor is it on your server, but perhaps you are using some form of internet security software which is specifically blocking you from taking these actions. Turn off that security software, if you have any.
Do you use a proxy to access the internet? Check the proxy settings. Maybe you have a proxy in between you and your web server which is interfering.
Are you on WiFi? Try wired. Try plugging into a different location to access it from elsewhere. Determine whether it is that computer or the location on the internet where that computer is.