The vote count for Ritchie drops from 37,365,173 at 4pm in CA (which would be 7pm in DC) to 34,091,976 at 7:15pm in the next scene which is in DC.
When Will is standing outside wishing for rain, the pavement is clearly wet from previous takes/tests.
When President Barlett enters the bullpen with the other staffers, you can clear hear the reporter on the television announce that Delaware and its three electoral votes are called for Governor Barlett, he is the President, not the Governor. The announcement was from the previous season when they replayed the election night from when Jed was first elected.
Josh is seen voting in person in D.C. at the beginning of the episode, but in episode 4.4, "Red Mass", he says that he would vote for Senator Stackhouse if he was on the ballot in Connecticut, indicating that he votes absentee in his home state. (We also know that he couldn't, since he has no address in Connecticut as his mother moved to Florida).
Donna spends the day trying to get someone to change their vote to the president to makeup for her accidentally voting for Richie. She claims that her vote in Wisconsin is more valuable because the state is in play. In season 3 episode 13 "The two Bartletts" Donna tries to get out of jury duty the next day. Donna would need to be a resident of DC to be selected for jury duty, in turn she would vote in D.C. Not Wisconsin.
During the day, Charlie tells Anthony and Orlando that he's going to go vote at 8:30. Polls in Washington DC close at 8:00pm.
The light and sound of the lightning shouldn't match. Light travel faster than sound, common mistake in movies and TV shows.
Rain is mentioned as a deterrent for people going out to voting places. At one point, Josh Lyman bemoans that "It's raining in Oregon." That actually wouldn't matter since Oregon has been voting exclusively by mail-in ballots since 1998 (when Oregon Ballot Measure 60 was passed).
Sam compares his agreeing to run for congress to the Secretary of Agriculture saying that he would be "ready to assume the Presidency should the 18 who come before me die." Secretary of Agriculture is actually 9th in order of succession.