This question may not previously have occurred to you, but now that it has, you probably welcome the following information:
(1) They would explode. Oxygen is not necessary. Fireworks have an oxidizer, and a reductant. (Sorry can't elaborate.)
(2) The picture shows fireworks being let off in the near vicinity of the Earth and the shards arcing back down as in any normal fireworks display. But suppose you went a long way off into inter-planetary space before letting your firework off, then it would be different. The shards would continue whizzing outwards pretty much in a straight line forever, faster than on Earth, without slowing. This is because of no atmosphere and little gravity. The only factor that would inhibit this effect would be the gravitational attraction of Jupiter or the Sun. Or some other planet, if nearby. The nearer, the more pronounced an arc there would be towards it.
(3) The colour would be disappointing. The reaction that imparts a rocket’s metals and metal salts with enough energy to display pretty colours requires oxygen. Unless your fireworks were specially designed for bursting in space, their colours would quickly fizzle out.
(4) The burst would be completely silent. No atmosphere means no medium to propagate sound waves.
(5) And by the way you would need to consider carefully how to light your firework. A match, I'm sure, does need oxygen, so my advice would be to provide yourself with a glowing electric filament.
Stefan Bossmann, a chemist and rocket enthusiast at Kansas State University, supplied the foregoing to popsci.com a popular science website. Though he forgot to mention points (2) and (5).
