Shuttles bring oxygen with them, because they have to burn their fuel to propel themselves through space.
It is because they carry liquid oxygen that they are so incredibly dangerous.
But really, fire is just an oxidation/reduction reaction.
All you really need are an oxidizer, a reducer, and heat.
Oxygen gas in the air serves as the oxidizer for most fires, and carbohydrates (lignin, cellulose, oil, petroleum, coal, etc) serve as the reducer in most fires you see.
But some rockets use nitrogen dioxide as the oxidizer instead of oxygen.
You could, theoretically, use hydrogen gas and chlorine gas to make a flame in the absence of oxygen - but oxygen is the only common and abundant oxidizer, since it is created as a byproduct of photosynthesis in plants and cyanobacteria.