I am working on a project that requires me to build what is essentially a camera monopod,
You'll have to make a very stable stand for it. More important than any number of snaps.
but with a head that rotates 360 degrees (in 13 increments) and vertically 180 degrees (in 11 increments).
Sounds like quite an overlap for stitching?
360°/12=30°each step and if you really do mean 360°/13, it's ~27.7°/step.
Or are you perhaps planning on using a short tele lens?
This new device will be an improvement over an existing design, which uses servo motors (would stepper motors be better?). Problem with that device is that its made of wood and the movements are very inaccurate.
It's not the material that determines precision.
The best way to get precise steps is to use position feedback and that can be used with a DC motor as well as a stepper.
The object is to take 100+ pictures which are then put together in a panorama-like picture.
You can do that with much less photos, so why the large amount?
Money is not a problem, I should be able to afford the needed parts.
If that's so, I just saw a very nice gimbal mount that could be converted, but you could build a fine system yourself, if you are into mechanics and know how to use tools.
Is this for a regular size (D)SLR or what camera are you planning to use?