I don't have sensitive armpits as far as I know and I have no friggin' clue what my deodorant has in it, per se, so you'll need to check that in a shop or look it up (I'm at work), but I used to swear by Vaseline Intensive Care unscented deodorant. It's a spray and it smells rotten (well, weird anyway, and not like anything like 'flowers' or 'vanilla' or whatever) in its own right, but on you I find that it basically smells like nothing at all; it really just cancels out sweatiness and the smell of unwashed (or washed-this-morning-but-been-doing-stuff-all-day) person. I would assume from it being Vaseline Intensive Care that it's reasonably hypoallergenic but I'll leave you to look that up for yourself!

However, my wife hates spray deodorant, hates the smell of aerosol and REALLY hates VIC deodorant smell, so I've started using hers instead which is the original unscented Dove stick deoderant. It does not offend me, it does its job, and I've never noticed it leave a residue particularly. What it does do that I hate is make my armpits feel damp whereas I like to feel dry (hence spray deodorant), but I got used to it in time.