The translation given is "How do we get to the airport?", but since 怎么 can also mean "why", can this sentence also be translated as "Why are we going to the airport?"
I'm probably missing something which makes this a silly question.
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I am a native Chinese speaker. This sentence is not ambiguous to me at all. Here the "怎么" clearly means "how" to me.
But if you switch "要" and "怎么"，then the sentence will be like:
Here "怎么" means "why". This sentence sounds to me like that you have a guest coming by train but your boss asks you to go to the airport to pick him up.
Hope this helps.
The position of 要 is very important to understand this sentence.
我们要怎么去机场？--> We 要(will) 怎么(how) to going to the airport? -We already decided that we will go to the airport. So "怎么" can only be explained as how.
我们怎么要去机场？--> We 怎么(why) 要(will) go to the airport? -"怎么" questions the willingness of going to the airport.
But when it comes to 我们怎么去机场？ You need contexts to understand the meaning. Or for oral language you can distinguish them by the stress: stress on 怎么 itself means how, on airport means why.
I think some are confused between making sentence with "怎么" and "怎么样".
"怎么" should means "why" and "怎么样" should means "how". The structuring of the sentence is thus very important, else it would be a "broken sentence".
怎么我们去机场？ - Why do we go to airport?
我们怎么去机场？ - Might sound correct as "How do we go to the airport?" but in actual fact it should be structure as "我们怎么样去机场？"
This is because of fast pronouncation of this sentence, which makes it sound like "我们怎么去机场？" or "我们怎样去机场？". Thus in a long run, people tend to say so.
The ambiguity that I see in the expression 我们要怎么去机场？is to do with what it is that the person is asking for, and normally this would be revealed in the context of the conversation.
But on the basis of the text by itself, I think it still means "How do we/How are we going to get to the airport?"
However, it is not so clear whether the person is wanting to know the path/direction or if they want to know the means by which you'll get to the airport (e.g. by car or train).