How much storm damage did it receive
Recently ???? or in the past?
When are you going? The Cinque Terre region (the 5 villages) didn't get a lot of damage at all although a lot of the trails are currently closed. That doesn't mean that more of them won't be open later in the season. That's not even an issue unless you intend to hike. Most of the damage was elsewhere in Liguria.
See this:
http://www.parconazionale5terre.it/Esentieri-outdoor.php
Two legs of the popular Sentiero Azzurro - or Blue Trail - have been closed from some time between Corniglia>Manarola and Manarola>Riomaggiore and are not expected to open during the 2019 season.
(edited a typo)