I am a US citizen with dual nationality (Antigua and Barbuda) living in the US Virgin Islands, can the USVI Board of Election Office make laws saying it is against VI election laws for me to vote in elections in Antigua and Barbuda if I vote in the Us Virgin Islands? Are there any US Federal Laws which precludes me from voting in both places?

Yes they can.

I know this person who was born in the USA but lived most of her life in Canada.

Years ago (1970s/1980s), this person was told that if she ever voted in a Canadian election, she would lose her American citizenship.

However, the Americans have since rescinded that law and therefore, she now votes in both US and Canadian elections.

But yes, a country can make laws like that.

To be a citizen of France, you can not be the Commander in Chief of another nation. So when Michelle Jean became governor general of Canada, she gave up her French citizenship.