To be held accountable for some harm done is to be required to pay the costs of making the injured party "whole", for example, paying for repairs to damaged property or paying the medical bills for injured persons, plus possibly paying some punitive fines above and beyond those incidental costs.
In your sentence, the main clause is cast in the passive voice:
A large and expensive class action lawsuit was filed [by someone]
A law firm filed a class action suit on behalf of the injured parties.
And the purpose of the suit is expressed by a (subordinate) infinitive clause:
... to hold accountable the party responsible for the disaster.
the party responsible for the disaster is the object of the verb in that clause. That party will be held accountable.