Designated competent authority(ies) to issue an apostille in Ecuador:

Director General de Legalizaciones, Ministerio de Relaciones Exteriores

Basic requirements:

Before of the apostille or legalization, the documents must be CERTIFIED OR LEGALIZED FOR THE OUTSIDE WITH PHYSICAL SIGNATURE by the entities that issue, being these originals or copies.

You can apostille by your own or a through third person, does not require authorization.