One of the most obvious external conflicts in The Wednesday Wars by Gary Schmidt is the Vietnam War, which serves at the backdrop for the entire novel. Characters lose loved ones (or, like Mrs. Baker, have loved ones who are deemed Missing in Action) and have philosophical differences about the war, like Heather Hoodhood and her father.
In a more personal sense to the characters in this novel, the conflict between seventh-grader Holling Hoodhood and Doug Sweiteck’s brother is a consistent external conflict throughout the story. Holling says this about the older bully:
I think something must happen to you when you get into eighth grade. Like the Doug Swieteck's Brother Gene switches on and you become a jerk.
The older boy is a bully and constantly torments Holling, though Holling does get in a few good jabs of his own. Doug Sweiteck’s brother rather threatens Holling into playing soccer at recess one day, and Holling ends up tripping the eighth-grader (who smashes his head into the goal and suffers a concussion). Another time, Holling is able to pelt Doug Sweiteck's brother (who never has another name) in the face with a snowball without the older boy knowing for sure that it was him. In between, however, the older boy torments Holling every time he can, including the incident where Doug Sweiteck's brother posts colorful but embarrassing photos of Holling in his Ariel costume (yellow tights with white feathers on the rear) all over the school for days.
These two external conflicts are a consistent presence throughout the novel.