How would an election work if some states are occupied or are combat zones?

It's not clear. The United States territories which were partially occupied in war but never parts of the state. The exception was during the Civil War where large numbers of States had seceded. In the Civil War, the seceded states did not take part in national elections. They did not have representatives in Congress and did not have electoral votes in the 1864 presidential race, the only one conducted during the war. Local elections were still held where possible. The question of how to readmit the seceded States was a major one for lawmakers during and after the war. If the US were partially invaded by some foreign power, there would need to be some sort of discussion of how to handle elections. Currently, elections generally happen anyway even if there are large numbers of refugees from an area. However, if an entire state were occupied it's an open question of what they would do

I do not think that most states have contingency plans for such situations and the nuts and bolts of election law is a matter of state law.

There are two separate issues. First, any "refugee" would have to decide whether they intended to stay in their new state (and register to vote there) or wanted to keep on voting in their original state in the hopes of returning after the war. Second, the states being invaded would have to decide what mechanisms would be set in place to cope with the situation.

In theory, states that are occupied could set up a legislature and government "in exile" (e.g., before fleeing Austin, the Texas legislature adopts a resolution to reconvene in Davenport). They would then have the voting records that would permit absentee voting (at least by those who fled the fighting). At the very least, such a legislature in exile could reclaim the right to select electors and give the governor the ability to fill Senate vacancies (caused by the inability to hold regular elections) until circumstances permitted a special election.

The closest precedent to this situation was during the Civil War when a rump group of Virginia legislators met in the part of the state under Union control. Congress treated this rump group as the legal government of Virginia (since the remainder was in an illegal rebellion and had thereby forfeited their offices under the Constitution), allowing it to consent on the part of Virginia to the loyalist part of Virginia becoming a new state (West Virginia).

They'd gather refugees and establish make-shift polling places as best they could in order to get as many votes possible out of those states to combine with the rest of the country and keep democracy flowing.