Each state has a Board of Elections that determines and communicates local election voting times. The Election Assistance Commission provides contact information for each state for easy reference at eac.gov.
The dates and times of local elections vary by state and also vary within each state from municipality to municipality. Contacting the local Board of Elections is the best way to get the most accurate information for a specific election happening in a specific town or county. This information is usually available on the website for the state, county or municipality. This is also the best resource when looking for local information for a federal election.