Can botox in the armpit really eliminate odor?

Botox can help underarm odor or bromohydrosis.  The injections are usually performed every 6 months and may be covered by some insurance plans.  Another option that has longer lasting results and may be of use if the Miradry procedure which destroys the sweat glands with microwaves.

Botox is an FDA approved treatment for excessive underarm sweating (axillary hyperhidrosis). Diminishing underarm sweating will in turn help improve underarm odor. 
Botox works to block nerve signals to nearby sweat glands that would normally tell the glands to release sweat. Without these signals your glands remain dry! The effects of Botox begin around day 3-5 but reach will reach full effect by day 14. Botox treatments to this area are temporary lasting approximately 6 months. 
I've created a video that reviews treating underarm sweating with Botox. I hope this information is helpful in your research!

Yes, for up to 6 months. It does so by reducing eccrine sweat which promotes bacterial growth (added to by heat and occlusion). It may also reduce odor through suppression of sweat through apocrine glands (attached to hair follicles). This sweat does contain lipids and proteins which themselves may cause odor. Botox doesn't kill any germs...it just reduces the sweat that is their breeding environment.