Breakdown of Dzieci składają dziadkowi życzenia.
Questions & Answers about Dzieci składają dziadkowi życzenia.
Dziadek is the basic dictionary form (nominative), used mainly for the subject of the sentence.
In this sentence, dziadkowi is not the subject; it is the person to whom the wishes are given.
Polish marks this “to someone” role with the dative case.
Dziadek (nominative) → dziadkowi (dative, singular, masculine).
So you use dziadkowi because he is the indirect object: to the grandfather.