Banned from a website. Need to use for work. Please help.

Question

I was recently banned from a website. I didn't do anything wrong and they won't tell me why I have been banned. My husband had an account on the website and as soon as they closed my account, they also closed his. We don't know if someone else used out IP address on this site and violated their rules or if we unknowingly violated their rules ourselves; they won't tell us.
The problem is that we need to be able to access this website for our work. We work from home and it's essential. We weren't sure if they had banned us by IP address, so we had a friend create an account. The account is legit and legal. He then gave us the username and password. We accessed the account from our computer at home and the account was almost immediately shut down. The friend was able to straighten things out with the company and get the account set back up, but we cannot use our computer at home to access this site. It's obvious that any access by this IP address will result in ANY account being closed.
We have a PC desktop, a MacBook laptop, and a wireless connection for the laptop (not sure how much info is needed, so I'm trying to be thorough).
I'm not going to ask someone to help me hack a site or anything. And after hours of research, I think we understand enough about proxy servers to use one of those effectively. I also know that I can call my ISP and request an IP address change (which may or may not be honored).
However, I have read other message boards and learned about cookies that websites send to your computer to identify you. I understand that most of those cookies can be cleared out by simply clearing the info from the browsing history (including cache, cookies, etc.) But I also read that they sometimes send files that are buried deeper in the computer. So here's my question...
How can I erase EVERYTHING that this website might have sent to my computer to be able to identify it? I must use the laptop to access the website. I'll work on the IP address issue on my own. But we can't use the desktop to access the site for reasons that are irrelevant and the explanation would be even more long-winded than what I've already typed here. We have to use the laptop. And I need to be ABSOLUTELY certain that the site can't identify the computer. I won't try to use my friend's account again until I'm certain that it won't be closed.
Let me reiterate this for those who are concerned: the friend has give permission for us to use his account, we have done nothing illegal on the website we've been blocked from, nor do we plan to do anything illegal on the website. We have done nothing illegal in any way, and I want LEGAL solutions to this problem. If we are not able to access this website, we will suffer financially in a severe way and are desperate to make sure that we can clear out cookies or whatever other tracking info has been put into the MacBook.
Can someone please help us?

Answer 1

Thank you for the info on onyx. I will try that. I actually took my computer to the Apple store and someone there "cleaned" it out for me.
As for why we can't resolve the issue with the website, we tried to talk to them about it but they just cut us off and said they weren't going to discuss it and that they wouldn't reopen the account.
We have one account as employees and the other accounts (the ones that were closed) were customer accounts. The two departments are separate, and we do not want to bring our account to their attention again and risk them connecting the customer accounts with the employee account and then closing the employee account as well.
We were using the customer account to monitor the employee account. Quality of video and sound, that kind of thing. Nothing fishy there. But since the guy on the phone didn't make the connection that it was our own account we were monitoring, he shut us down. Customers do the same thing to us and basically rip us off for thousands of dollars every week, and the company never gives those customers any trouble about it. So it's unbelievably frustrating. Also, we figured out how to do this by watching the activity of the other employees, since there's no "training" and we all learn from one another by watching. We don't know what would happen if someone there made the connection... whether they would just say "oops" and reopen the customer accounts, or whether they would be irritated about us using the customer accounts in that way and just shut us down completely. We'd rather not find out.
As of now, we're using a friend's account to monitor our video and sound quality. But we're still using the same computer. That's why it needed to be "cleaned."