The sites mentioned (like personal online banking) by others probably don't transact billions of dollars a day. But at least the examples provided show that web sites can provide a fairly reliable service. They do go wrong sometimes, just as your fax machine might, but by and large they are online well over 99% of the time.
There are plenty of examples which do transact very large $ amounts over the Internet. They do have to pay to get this reliability, but I was responding to a comment which suggested that web applications couldn't be reliable enough for NOTAMS. And I personally know of finance sites that transact very large $ amounts who pay 5 figure sums per year.
Lastly, there are plenty of e-commerce sites around whose hosting bill is a few thousand pounds per year that enjoy > 99% uptime.
It is true that the AIS site is not commercial - you won't go elsewhere in disgust. And it is safety critical. But, as others have said, simple text based NOTAMS could be made available from a number of servers for a tiny cost per year.
The US standards mentioned earlier are well above 99% uptime, but if the AIS site was available and responsive (leaving aside for now the question of actually being able to get the data you need, just for the sake of this discussion of hosting) 99 times out of 100 and you had to use a simplified alternative for those other times, you'd probably be reasonably happy.
For those that care, AIS is now at 91% for the last 4 days. The big hit was a 7+ hour outage, which makes me think that they don't monitor it or can't resolve problems remotely.