I believe WICEN have done HF internet before from my understanding. However, due to instablity in the upper atmosphere, they have decided to switch from HF to VHF.
There is no doubt the ham community can set up internet type links especially via VHF, packet radio and repeaters, ie no sattelites, no land lines, no cable and in very short time. The infrastucture is in place in Australia, known as WICEN, and this organisation regularly provides emergency communicationss when everything else is down. A link with this community is a must for Troubled Times, and with the purpose fully explained expect real support and some amazing knowhow, especially with the "Heathrobinson" emergency makeshift gear.
It is possible to build a packet radio to patch into a VHF transmitter and computer. Via ham repeaters "free" computer links and bulletin boards were in regular use long before anyone had heard of the internet. Today hams have taken this to an art form worthy of better understanding. A survival site should have a tower and a good VHF antenna that can work many stations simplex and many more via repeater. Web sites such as Iphone are available with pertinent information.
The way radio frequency propagation works is that with HF (high frequency, i.e. from 1 meg to 30 meg) most long distance short wave ham radio takes advantage of bouncing such radio frequencies off the Ionosphere. This is used for best long distance results, and a great favorite of hams, but these frequencies would not be useful with upper atmosphere interference. But VHF and UHF are much higher frequencies and do not bounce off the ionosphere - they go straight through it! Thus they are not so affected by disturbances in the upper atmosphere. My best guess is there might be some problems, but VHF would get through and UHF is even better.
A packet terminal node connector sits between the transmitter and the computer, and does not know what frequencies the operator is using any way. It is being used right now world wide with great results by tens of thousands of hams.