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As it happens, I have got an old laptop here which runs Vista. I use it for running some old bits of software. So I did a quick test and this is what I found.
The NYA website does work with Vista but it depends which browser you use. It works fine with Firefox and Internet Explorer 9. With Google Chrome you get the error message. With SeaMonkey, nothing happens at all. No website or message.
I don't have any other browsers installed on the machine.
I strongly suspect that there is some bit of code on the website which requires an up to date browser. Google Chrome no longer receives updates for Vista machines so it might be missing something which that particular page needs.
Not all web browsers are the same and it is not all that unusual to find websites which particular browsers will not display. I had this problem on a page in the Ludham Archive website and I had to put in a bit of code to put a certain browser into compatibility mode. In the end I changed the code on the page altogether so it worked.
As Vista is long out of support and gets no updates it will gradually slip behind and eventually will not be able to support some new functionality.
Maybe it is time to move on, but you could try downloading Firefox as this is working for me. Windows 10 makes Vista look like a trip into the dark ages now.
You could convert old slow machines to Linux but that might just give you a different set of issues.