6 weeks post Botox, can half your face become numb if you get injected in the axillae?

Nope, this is unrelated to your Botox. I would recommend following up with a Neurologist for an evaluation. Best wishes.

Botox used for hyperhidrosis acts locally on the sweat glands, to prevent the release of Acetylcholine and stop the sweat glands from producing sweat because of "medical denervation". Numbness on the side of your face is not related and could be something neurological such as Bell's palsy, post viral, or even a TIA. I would recommend you see a neurologist or visit an emergency room if this is something acute! 

This is not an expected outcome of any botox. There is no sensory loss with Botox so i you face has become numb especially 1/2 of your face makes be believe this is neurological. Ill advise you to see your primary if not a neurologist as soon as possible, like today!