Yes, the sentence is perfectly proper English.
It is possible to construe two different meanings from the sentence, yes; but the second meaning is extremely unlikely. I had to read the sentence four times after reading your question before I realised what the second interpretation would be. It would be a very strange question to ask. (It’s already a bit odd as it is—what does not wearing shoes have to do with making drinks?)
In fact, there are at least four possible ways you could construe this question—each more unlikely than the next.
It is only ‘bad style’ (if you can even call it that) to use structures that are highly ambiguous and remain so even in context; this is definitely not that. Even without context, the risk of ambiguity is so small here that there is no need whatsoever to worry.
Basically, the options I can think of are:
Do you want [someone who has taken off their shoes] [to prepare your drink without washing their hands]?
= If someone is going to prepare your drinks without washing their hands, do you want it to be someone who has taken off their shoes?
Do you want [someone who has taken off their shoes] [to prepare your drink] [without washing their hands]?
= If you’re not going to wash the hands of the person who’s preparing your drink, would you want them to have taken off their shoes? (Bizarre question.)
Do you want [someone who has taken off their shoes to prepare your drink] [without washing their hands]?
= If there is someone who has taken off their shoes in order to prepare your drink, do you want them without washing their hands? (Utterly grotesque question.)
Do you want [someone who has taken off their shoes [to prepare your drink without washing their hands]]?
= If there is someone who has taken off their shoes in order to prepare your drink with dirty hands, do you want them? (Downright outré question.)
— And I’m sure you could think of even more mind-boggling ways of interpreting the question.
The point is: everything is ambiguous. Every single sentence you utter in a day can possibly be misconstrued to mean something else than what you intended for it to mean. This is very rarely a problem, however, because people don’t usually spend their time trying to find obscure and bizarre meanings in what other people are saying to them—they don’t have the time, because the conversation has already moved on and they need to come up with an answer as well.