Are you referring to Star Smashers of the Galaxy Rangers? This is not a short story, but rather a 1973 novel by Harry Harrison. Other than the fact that it is a novel, Star Smashers fits what you're looking for, from the cheese-powered ship, to the war, to the humorous tone.
The novel revolves around two college science students who develop a faster-than-light drive powered by cheese ("Cheddite"). They eventually end up in the middle of an intergalactic war.
The wild, galaxy-hopping adventures of brash young scientists Jerry Courtenay and Chuck van Chider are at the core of this classic space opera. When the two college students develop a faster-than-light space drive in their homemade workshed, they decide to sneak it aboard their football team's airplane as a prank. The boyish plan backfires, however, and the boys find themselves, along with their crush Sally and the seemingly loveable school caretaker, Old John, hurtling through the solar system towards Titan — an icy moon of Saturn inhabited by hideous ice creatures. Titan and the 20th century are only square one as the foursome becomes embroiled in a vast, intergalactic, century-jumping battle.
From the novel's Wikipedia entry:
It is a parody of the space opera genre and in particular, the Lensman and Skylark series of E. E. "Doc" Smith. It also includes an homage to Larry Niven's Ringworld.
It is about two college students, Chuck van Chider and his friend Jerry Courtenay, who accidentally invent a device that can transport them through space, powered by a substance called "Cheddite", which is created by irradiating Cheddar cheese.
Note: In case you are curious, I found this book by searching for specific cheeses. ("Spaceship" combined with "cheese" wasn't pulling up anything that fit your description. However, "cheddar" did.)