Political donations are a matter of public record, so it is possible one could get in trouble over that: Mozilla CEO Brendan Eich famously resigned after controversy around his donation to a campaign opposing gay marriage. But this is a rare situation, since the CEO of a nonprofit is kind of a political figure; it's not something an ordinary employee would worry about.
In the 36 states where voter rolls are also public record, some batshit-paranoid executive could theoretically find out which employees didn't vote and give them a hard time about it, but this would be completely unheard-of and possibly illegal. Many US citizens don't bother to vote; nobody cares.
This is tangential, but in Sweden, voting was not secret until 2019:
Long-standing Swedish election policy of always displaying the ballot papers for voters to select in public, making it impossible for many voters to vote secretly, has been criticised as undemocratic and is arguably in blatant contravention of Protocol 1, Article 3 of the European Convention on Human Rights [..] In 2014 a German citizen, Christian Dworeck, reported this lack of secrecy in Swedish voting to the European Commission and from 2019 ballot papers are selected behind a screen.
2. Voting status is not a protected class. If they want to fire you over it, they can. This seems absurd though.
I’d suggest you find another job if you think you’d be “into trouble” by voting a certain way or not voting.
Perhaps just keep your personal opinions to yourself if you think you’ll be ostracized… but I would not want to work in such a place.