Just means to cast your vote. (with a ballot)
Registration is the word you seek.
im not sure about a 7 letter word, but i do know i 8 letter word that starts with s and means the right to vote, the word is... sufferage.
personal or discreet
disenfranchise
The Latin equivalent of the English phrase 'the right to vote' is Ius suffagium inire. In the word-by-word translation, the noun 'ius' means 'right'. The noun 'suffragium inire' means 'to vote'. The English word 'suffrage' comes from the Latin noun 'suffragium'.
enfranchise
Vote comes from the Latin word votum- which means wish. When you vote, you express your wish of who should be elected.
Women who were seeking the right to vote were known as Suffragettes. This word is derived from suffrage, which means the right to vote.
Polls are set up in numerous locations all across cities, states, and nations in which people may vote. If you mean to ask for the word that means "the right to vote", then the word you are looking for is "suffrage".
Eligible means you are old enough to do something e.g. to vote. In the specific case of voting, it means you are allowed to vote if you register to do so.
veto