Synthesis of nitrocellulose
The sulfuric acid acts by protonating nitric acid to yield nitronium and hydronium ions. 


Electrophilic nitration reactions like the production of nitrocellulose require sulfuric acid because of the dehydrative properties of sulfuric acid. Sulfuric acid is a non-nucleophilic acid - all it can do is mediate proton transfers and dehydrate things (sulfuric acid has a large negative enthalpy change of mixing with water). Thus,$\ce{NO2+}$ is free to react with whatever you want it to.

$$1)\ \ \  \ce{H2SO4 + HNO3 &lt;=&gt; HSO4- + H2NO3+} $$
$$2)\ \ \ \ce{H2NO3+ &lt;=&gt; H2O + NO2+} $$
$$\text{overall)    }\ \ \ce{HNO3 + H2SO4 &lt;=&gt; H2O + NO2+ +HSO4-}$$

Hydrochloric acid and nitric acid produce aqua regia. Hydrochloric acid is a nucleophilic acid, and thus the decomposition products after proton transfer to nitric acid are different. The chloride ion is a good nucleophile, and reacts with the nitronium ion, eventually producing $\ce{NO2}(g)$ and $\ce{Cl2}(g)$. There is a big difference between $\ce{NO2+}$ and $\ce{NO2}$. 

$$1) \ \ \ \ce{HCl + HNO3 &lt;=&gt; Cl- + H2NO3+}$$
$$2) \ \ \ \ce{H2NO3+ &lt;=&gt; H2O + NO2+} $$
$$3) \ \ \ \ce{2NO2+ + 2Cl- &lt;=&gt; 2NO2 + Cl2}$$
$$\text{overall)    }\ \ \ce{2HCl + 2HNO3 &lt;=&gt; 2H2O + 2NO2 + Cl2}$$


  So nitrate salts can be used instead of nitric acid because all you really need are the nitrate ions. 


Yes. However, starting with nitric acid shifts the equilibrium towards nitronium a little.


  Also, wouldn't very small amounts of nitronium ion be present without the sulfuric acid anyway? Could one prepare nitrocellulose using only nitric acid if it were soaked for a much longer time?


Maybe, but not enough to be useful. Nitric acid is not available pure. It is only stable to a maximum concentration of 70-75% by weight (remainder water). The other reason that sulfuric acid must be used is that it is available "pure" (or at least 98% by weight) and it is so hydrophilic so as to sequester the water. Hydrochloric acid has a max concentration of 35-40% by weight remainder water. All the extra water gets in the way and shifts the equilibrium in the opposite direction (the reaction produces water).