Aww, how cute -- they want to be just like mum! I rang my GP and this is what she had to say:

Deodorant isn't designed for little kids -- it has chemicals and compounds in it that are meant for more mature skin, and as they are highly concentrated, it's not a good idea to use deodorant on very young children. Deodorant is for perspiration and odour control, which become issues along with changing hormones, and is formulated to address those needs in older children and adults. Kids don't need to start using deodorant until they have noticeable body odour, which coincides with puberty for most children.

That said, what if you get an old deodorant container (the roll-on kind) and fill it up with water or baby powder, and let them "pretend" with that? Or an empty stick-deodorant container with something cut to fit the empty hole -- soap, or a sponge, or something non-toxic for them to swipe under their arms. It's got to be better for their skin than the real thing.