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!

Thank you for the question. Based on your description of the history of events, I doubt there is neurotoxin that transferred or migrated from your axilla to your facial nerves.  There is a possibility the Botox was injected too deep in the axilla causing some migration to adjacent tissues, but, again, the likelihood of neurotoxin migration to such a distant location and 1.5 months later is very unlikely.  You should probably have an appointment with your primary care to discuss this matter and discuss your symptoms with the provider who injected you with neurotoxin.

Sounds like unrelated issues.
Have a neurologist help you with the answer to your problem. Consider migraines in your differential.
Best. Dr Brecht