First of all, it is important to note that you have left out a very important word from this quote. The word is “never.” The correct phrase from the Declaration of Sentiments is
He has never permitted her to exercise her inalienable right to the elective franchise.
The Declaration of Sentiments was the statement that was put out by the people (mostly women) who met at the Seneca Falls Convention in 1848. This is seen as the beginning of the movement for women’s rights. The Declaration of Sentiments was explicitly meant to mirror the way that the Declaration of Independence had been worded in 1776.
In the Declaration of Sentiments, the writers are listing a number of injustices that have been done to women by men. When they say “he” they do not refer to any specific man. Instead, they are referring to men in general. This mirrors how Thomas Jefferson wrote a long list of ways in which, he said, King George III had abused the American colonists.
The phrase that you quote means that men have never allowed women to vote. The “elective franchise” is the ability to vote. An “inalienable right” is one that cannot be legitimately taken away. Thus, this phrase is saying that women have a right to vote but men have not allowed them to do so.