They dreamed that someday they would gain the right to vote.

Literal

They [topic-は] someday [topic-は] voting-rights [subject-が] obtained [probably-だろう] thing [object-を] dreamed.

選挙権が得られるだろう事を夢見た — 選挙権 means 'suffrage, right to vote.' 夢見る means 'to dream of.' A historical sentence about women's suffrage.