I don't believe elections were ever mentioned in Trek, but I think its safe to assume that the Federation is a democracy, where all individuals (once reaching an adult age in accordance with their species development and customs) would be given a vote.
Only makes sense to me, otherwise you go down the route of a dictatorship, which just isn't Trek.
Commander Austin Harris, First Officer, Deep Space Nine (by FltCpt. Bossco)
8.01 - Darkest Before Dawn
(Chapter 8 added, 12/09/2015)