What is the oxidation mechanism of gunpowder?
$\mathrm{KNO_3(s)}$ can oxidize charcoal without sulfur. It needs a "hot" primer through to get the reaction started.  Sulfur's role is to lessen the ignition temperature. 

See section in Wikipedia article.

Since the reaction isn't complete and has a number of possible products bthere isn't an exact chemical formula for black gunpowder burning. The Wikipedia article lists several approximations to black gunpowder burning. What is supposedly the best follows. 


  
    Although charcoal's chemical formula varies, it can be best summed up by its empirical formula: $\ce{C7H4O}$. Therefore, an even more accurate equation of the decomposition of regular black powder with the use of sulfur can be described as:
    $\ce{KNO3 + C7H4O + 2S → K2CO3 + K2SO4 + K2S + 4CO2 + 2CO + 2H2O + 3N2}$