I think this is a good idea! Maybe even allow for internet access on the computers, for a random chance to try and gain a new skill level (or some skill experience), learn a location of interest, get a map, learn a new recipe or get music/photos/morale boost, or morale penalty because of bad connection, reading about the apocalypse on the internet or something.
In a world where there’s cybernetic implants, atomic vehicles, power armor and laser weapons, I can’t see how some tech stuff like this would be unrealistic. Maybe make the chances for a positive event on the internet be very low - just have the character waste time and surf the internet (wasting time and batteries).
Maybe add different levels of internet access using different equipment (routers, etc.). Would be a nice future feature to have the possibility of researching stuff on a good network setup. Allow for the character to create new custom stuff (make some sort of in-game editor which writes to JSON files) and 3D print them.