Origins
The word 'vote' began its life as a prayer. It descends from Latin 'vōtum' (a vow, a wish, a solemn pledge to the gods), the past participle of 'vovēre' (to vow, to dedicate, to promise solemnly to a deity), from PIE *h₁wogʷh- (to speak solemnly, to vow). The transition from sacred vow to secular ballot is one of the more striking secularizations in the history of political vocabulary.
In Classical Latin, a 'vōtum' was a religious act -- a solemn promise made to a god, often conditional: 'If you grant me victory, I vow to build you a temple.' Roman temples, monuments, and public works were frequently the fulfillment of such vows. The word carried weight, gravity, and the force of sacred obligation. It was not a casual expression of preference but a binding commitment witnessed by the divine.
The word entered English in the 1530s directly from Latin, in the context of parliamentary and ecclesiastical proceedings. The shift from 'sacred vow' to 'formal expression of choice' may have been mediated by the practice of monastic voting, where members of religious orders cast 'vota' (vows/votes) on matters of governance -- a context where the religious and the procedural naturally merged.
French Influence
The doublet 'vow' entered English much earlier, in the thirteenth century, through Old French 'vou' (a vow, from Latin 'vōtum'). So English has two words from the same Latin source: 'vow' (the religious/personal promise, arriving early through French) and 'vote' (the political act, arriving later directly from Latin). The different arrival times and routes gave each word its distinct semantic territory.
The derivative 'devote' (from Latin 'dēvovēre,' to vow away, to consecrate, to dedicate) entered English in the sixteenth century. Its original sense was intensely religious -- to devote oneself was to make a sacred dedication. The weakened modern sense ('I devoted the afternoon to gardening') retains only a shadow of the original gravity. 'Devotion' preserves the religious sense more strongly, especially in the plural ('morning devotions,' meaning prayers). 'Votive' (from Latin 'vōtīvus,' of or pertaining to a vow) survives primarily in 'votive candle' -- a candle lit in fulfillment of a vow, one of the last living traces of the word's sacred origin.
The political history of voting is far older than the English word. Greek 'psēphos' (pebble) was used for votes because Athenians voted by placing pebbles in urns -- our word 'psephology' (the study of elections) preserves this. Latin 'suffrāgium' (the right to vote, a voting tablet) gave English 'suffrage.' The word 'ballot' comes from Italian 'ballotta' (small ball), because Venetians voted with small balls. Each word for voting in the Western tradition encodes a different technology: pebbles, tablets, balls, vows. The English word preserves the most solemn of these -- the vote as a sacred pledge.