Ok, I'll say this before I launch into the lecture you didn't ask for lol..... sorry I have a tendancy to do that!! But I'd be a lot more worried about, say, the lactic acid. I think this product is NOT benign or natural and seems like it might break down latex. If oil can do it, acid certainly can, right? I don't really know, this is just my speculation. Since they haven't done the research, they probably can't tell you anything (legally) about how they might interact with condoms. And since they haven't done the research, I wouldn't trust it, even if you find out that there's not oil. Oil's not the only thing you should worry about.
Ok, now for the lecture you didn't ask for lol.....
The vagina is a self cleaning organ. A healthy vagina does not need deodorant. Are you having problems with infections, or are you just self conscious about your natural scent? A slight smell is normal, but if it smells foul or strong, that's a sign of infection.
Umm.... did you think about looking for studies or trials to determine if this was safe to put in your vagina? That seems like it should be your first concern.... it doesn't sound very safe to me, actually it sounds like you will get infections from it.
But I don't know, if you decide to use it, let us know how it worked out.
I tried to find info on it online, but couldn't find anything reliable. It doesn't seem that it's FDA approved or that ANY research has been done on it. It seems like a gimmick to play on women's self-consciousness, at least in my opinion. Just like douches, that doctors now say are a BAD idea and totally not necessary.