Voting from abroad in Japanese elections
Japanese nationals living abroad can vote in national elections, through the overseas voter register. Registering is a separate act from the residence notification, although both are filed at the consulate.
The two things to file, and the order

How you vote once you are registered
Either in person at the consulate during the voting period it announces, or by post from Japan. The consulate publishes the dates for each election, and the in-person window is short.
If you move within the United States
Moving between consular districts means notifying the change. The districts are not intuitive: Nevada belongs to San Francisco, West Virginia to New York, and California and Connecticut are divided by ZIP code. A move of a few miles can change which consulate holds your file.
The residence notification, which does more than you think
The overseas residence notification, filed when you move abroad, is what connects you to the consulate: it is how the office reaches you in an emergency, how it counts the Japanese community in its district, and the basis for the voter registration.
It can be filed online through ORR net without visiting the counter, and updating it after a move takes minutes.
Which elections you can vote in
National elections: the House of Representatives, the House of Councillors, and national referendums. Local elections in Japan are not open to voters registered abroad.
The voting card is issued by your last municipality in Japan and posted to the address on your notification. An out-of-date address is the commonest reason a registered voter cannot vote.Voting in person at the consulate
Each office announces its voting period, and it is short: a handful of days, within counter hours. Offices that cover several states are a long drive for most of their district, which is why the postal route exists.
If you return to Japan
Registering as a resident in Japan again removes you from the overseas register. Moving back abroad later means registering again from scratch, with the same weeks of waiting for a new card.
