I was born and raised in Ohio [snip] I don't get to vote for a United States Senator from the State of Ohio this year.
But if you went, not to another American state, instead to work in another country and stayed a American citizen, you would be able to vote in Ohio state elections and in federal elections as well.
Well, let me put it this way
I'm on planet Washington (but born on planet West Germany), but I can't vote for a likely candidate for the next presidential term from my home planet. So I have to vote for (one) the candidate from planet Hawaii who moved to planet Illinois, or (two) the candidate from planet Michigan who moved to planet Massachusetts.
Both of these men are of species American, as I am, so I can vote for one of them
Even though I was born on planet West Germany, I don't vote for species German candidates. Legally I can't, because I'm species American.
As my parents (both from planet Washington) moved about the Federation during their military careers, they voted in planet Washington's many elections by absentee ballot. They did not vote as residents of whatever planet they simply found themselves on, even through they often lived there for years.
Even after both my parents exiting the military and we were living on planet Brazil for a year, they sent ballots to planet Washington..
Amanda is from planet Earth, and is species Human, she can vote "absentee" for species Human issues and candidates. Even if she were born on Vulcan she could vote Human. She could (if she wished) run for species Human political offices. For as long as she lives on Vulcan she would be a resident foreign species, not a "Vulcan."
It is possible that the Vulcans would let long term resident foreign species vote in local and regional elections (their choice), but as a "Earth-woman" she could vote in major Earth issues and affairs. Thing that effect her species. Just as Vulcans residing on Earth would make their voices know back home on major Vulcan issues.
What I'm suggesting is just like what we have in America today. Over 6 million Americans civilians live abroad, plus about another 0.3 million military. some will be outside for years, or perminately. They're all American citizens, they can vote in elections. Percentage wise, that would be like 140 million Humans living "off-world."
Look at how many people live inside America, without being citizens. Approximately 30 million, they don't vote in elections.
None of that precludes her from obtaining Vulcan citizenship and being represented on the Federation Council ...
None of that requires her to be legally a Vulcan person. As long as Amanda is selecting a representative, it makes no difference if that person lives on the same planet as she. The representative does need to represent Amanda's views and beliefs.