Of course, the main clause is third person plural present tense, with a dependent clause as the object, and surely you see the dependent clause (also third person plural present tense) with the implied "that" as the link and the predicate adjective "important", but I assume it is the predicate that is giving you trouble: "do think" is a compound predicate, with "do" being an emphasizer, not to be confused with a verb plus fragment (as "put up with").
So: They think = main subject and predicate
that = implied link, making the dependent clause (national elections are important) the object of the main clause
national elections = noun phrase subject (national is an adjective modifier) of dependent clause
are = predicate of dependent clause
important = predicate adjective