No offense, I find it troubling that you were cut off without warning, but you are using a computer that was introduced in August 2007 with an operating system that has been unsupported since July 2018. I'm not saying that your provider was right in doing what it did, but running on older hardware and software does come with risks. This is one of them.
Do you have a smartphone or tablet that is slightly newer and could perhaps enable access to the mail account? I suspect you're using this through your browser on your computer, but setting up a dedicated mail program on a computer or other device may be a better way to get access and control where your e-mails end up.
Edit: Disregard that last paragraph, I see that Tutanota is aimed at providing an encrypted service and only supports the online version or their own desktop client. They do clearly state that they only support
current versions of various browsers:
https://tutanota.com/faq#browser-support The Firefox 78.15esr version you're using is from 2021 as far as I can tell (we're up to version 111.0 on the Mac now), so there has been some leeway here, but I fear that you were always going to end up in this situation at some point. It is simply not possible to provide support for outdated software versions forever, next to the fact that as a company they may be putting their servers at risk by allowing a less-secure browser to access the system. I haven't looked at their terms of service, but there may be a line in their stating that you as a customer have a responsibility to keep your end as secure as possible. If that is the case, you may not have a very strong case against them. That's just my two cents.