I have tried the Magnesium Oil suggestion, when I went away for a few days and took an empty deodorant with me
I found that it worked perfectly as a deoderant.
However, it was not a comfortable experience (oily armpits anyone?) and actually seemed to chafe a little.
Of course, I may have used too much, or not enough, or whatever.
I have seen that Weleda do some wildly expensive aerosol-less sprays. I tried the rose one several years ago, but was disappointed in its effectiveness.
Lush also do a variety of different options, from powders to creams. When I tried a cream, I found it built up on the skin, and the powder version left white marks on clothing.
I suspect that all the above are vegan, but can't vouch for it - but I am sure you are very used to scrutinising the packaging with care!
Ed. For spelling