When do I use the hydronium ion in an acid-base equation?
In both your cases, only H+ actually participates in the reaction. Remember that HCl is a strong acid, and dissociates completely into H+ and Cl-. For example, consider the net ionic form of your second equation:

$\ce{C2H3O2- + HCL -&gt; HC2H3O2 + Cl-}$

$\ce{C2H3O2- + (H+) + Cl- -&gt; HC2H3O2 + Cl-}$

$\ce{C2H3O2- + H+ -&gt; HC2H3O2}$

For your first equation, also note that, similarly, although the entire H30+ molecule participates in the reaction, only the H+ is transferred. (Also, I think you meant NH4+ instead of NH3 as a product.) So:

$\ce{NH3 + (H3O)+ -&gt; NH4+ H2O}$ 

$\ce{NH3 + H+ -&gt; NH4+}$

So basically, they both use the same compound (H+). Usually water is the main producer of H+ (through self-ionization). You just need to consider factors such as the common ion effect that would make other compounds such as HCl the main producer of H+ instead of water.