The approach you need for deoderant depends on whether you have mainly self-produced musk, or if you get bacteria, yeast and fungus causing a stinky rot in the warm dark moistness. Some things that I have used to reduce microbial action and have heard of others using with fair success in reducing resulting odor:
Vinegar. Not as long lasting as some things, but keeps bacteria and fungus from blooming in the moist heat and causing a stink for hours. I just have to remember to keep the sprayer away from any razorburn...

This can be diluted to reduce discomfort of application, but I have sprayed or wiped on straight 5% with a great reduction in stench, and the added advantage of wiping out my athletes foot.
Colloidal Silver. Most of the advantages of vinegar, and more. Longer lasting, no mess or smell. Not as easy to buy or store, but a bit easier to make yourself...
Alcohol. Sort of like listerine for the underarms... Not as long lasting as the above. Listerine could also be used.
Other items can be used such as:
HOH or hydrogen peroxide. Tends to eat holes in the clothing and not as long lasting as the others.
Betadyne... This might be rather effective, but the resulting color may be rather socially inconvenient.