Preparation of ammonium acetate buffer for HPLC
to prepare a 100 mM ammonium acetate solution at pH 9, you’d first prepare a 100 mM ammonium acetate, and then add ammonium hydroxide dropwise until the desired pH was achieved.

Well, this procedure is good enough for a rough screening solution- call it a quick and dirty method. The main issue is that one would not prepare a solution of 100 mM ammonium acetate beforehand and add drops of ammonia to that. Whenever you need to prepare a buffer solution, you should know what volume is desired and what concentration is needed.
For HPLC, let us say you need 1000 mL of buffer containing 100 mM ammonium acetate. One would dissolve the required amount of ammonium acetate in about 900 mL water in a 1 L beaker, add ammonium hydroxide dropwise till you reach the desired pH, add more water till you are close to 950 mL. Check the pH again, and adjust as desired, and then quantitatively transfer the solution to a 1 L volumetric flask and then make up the volume to 1 L. This will be your analytically correct 100 mM ammonium acetate buffer.
Now, coming to the buffer equation. Again, for that you need to define the concentration and volume. What would ask yourself what should be concentration of ammonia should be present in 1 L of 0.1 M ammonium acetate solution such that the final pH is 9 solution.
You would use the pK$_a$ of ammonium ion and from the equation you would figure out the concentration of ammonia.
See example no. 2 here: Buffer calculation
And also study example 3. Ammonia buffers
Now the problem is that conc. ammonia is sold as w/w % solution. Using the density and percentage, you need to determine the molarity of ammonia solution as well to actually prepare a buffer.