What chemical reactions could ancient people use to propel projectiles?
The answer depends on what do you mean for ancient people. There are a lot of reactions that could produce gas for ejecting projectiles. The most common was the mixture of sulphur, potassium nitrate $\ce{KNO3}$ and carbon that is known as gun powder and has been reported to be used since the IX century in China.

If you want some less efficient and not used reaction,  you could think of using methane explosion extracted from some swamp. Or some reaction between calcium carbonate and acetic acid. Like this one between sodium hydrogen carbonate:

$$\ce{NaHCO3 + CH3COOH -&gt; NaCH3COO  + CO2 + H2O}$$