It is possible. More so for bunkers in Eurpoe to be connected with one another or with those in Asia, or North to South America. You just lay cables as we do now. or you could just lay them on the surface, if there is no-one left to steal them. If you are thinking of potentially laying thousands of miles of tunnels then this is not a trivial task, but threading cables down them once constructed is simple.
The less relatively easy bit may come when havig to traverse oceanic treaches, subdution zones or mid ocean ridges. But, with current technologies we run cabels between continents to supply the internet to the world.
So communtication would be fine, but shifting power over long distances leads to large percentage losses, so it is possible but expensive. And rather than DC transmission which requires lage conductors, you would want high voltage AC which requires large distances between cables to prevent arcing... this is why pylons hold the cabels so far apart.
You could also use short wave radio to communicate between bunkers bouncing signals of the ionosphere. Then they would not have to be physically connected to be able to exchange information. But bandwidth is low. If there were still satalites these could potentially be used to communicate.