Yes.
The "air" for combustion of the powder is contained in the molecules of the powder itself. Firearms will fire even underwater. Even if there were a "leak" in the cartridge, so there was a vacuum in the case, the powder would still burn, since it provides its own "air," or oxygen.
The "air" for the burning is in the form of the three oxygen atoms in the nitrate radical, -NO3, which is attached to the cellulose molecule or the glycerine molecule. (That's why they are called nitro-cellulose and nitro-glycerine.)
In simple terms, disturbing these "nitrated" molecules by means of the primer flash causes a rearrangement of the atoms so that the oxygen is free to combine with the carbon and hydrogen in the glycerine or cellulose to release a lot of heat energy. The resulting high-pressure hot gases push the bullet out of the gun.
The primer also contains its own oxygen, but is sensitive to the firing pin blow, which starts the primer composition burning --quite violently.
Smokeless powders are of two types --"single-based," which means they are nitrocellulose only, or "double-based," which means they are a combination of nitrocellulose and nitroglycerine. Ball powders are usually double-based.
If I recall correctly, the double-based Bullseye, which is about the fastest-burning powder you can buy for reloading, contains about 13% nitroglycerine and the rest is nitrocellulose and a small amount of stabilizing chemicals.
The older black powder also contained its own oxygen, but it was not combined directly with another molecule, as with smokeless powder. Its oxygen is contained in the nitrate in the separate saltpeter molecule, potassium nitrate (KNO3) or sodium nitrate (NaNO3). When heat is applied to black powder, the oxygen in the KNO3 or NaNO3 "comes loose" and will provide the oxygen to burn the sulphur and charcoal of the black powder.
The composition of black powder varies, but is basically
73% saltpeter (potassium or sodium nitrate)
15% sulphur
12% charcoal (carbon)
Other "oxidizers," such as ammonium nitrate, NH4NO3, are also used in black powder substitutes and explosives.
The -NO3 radical, containing the three oxygens, is found in most propellants and explosives It is very cheap.
Other radicals, such as the perchlorate and chlorate radicals, -ClO4 and -ClO3, operate in the same way and are also used in propellants and explosives.