Only one implication of the equivalence appearing in the question holds.

It is true that if each connected component of a poset $(P,\tau)$ can be endowed with a Priestley topology, then the same holds for the whole poset. There is a construction in the proof of Lemma 2.3 of this paper.

However, in the same paper in Lemma 2.4, a construction of a Priestley space $(P,\leq,\tau)$ is given such that one connected component of $(P,\leq)$ is not Priestley-topologizable.

Only one implication of the equivalence appearing in the question holds.

It is true that if each connected component of a poset $(P,\tau)$ can be endowed with a Priestley topology, then the same holds for the whole poset. There is a construction in the proof of Lemma 2.3 of this paper.

However, in the same paper in Lemma 2.4, a construction of a Priestley space $(P,\leq,\tau)$ is given such that one connected component of $(P,\leq)$ is not Priestley-topologizable.