People would dodge or divert a question for many reasons. It can be that they don't have an answer and they don't want to be perceived as unknowledgeable or they might hold a hidden agenda and wish to maintain it. 

Politicians are notoriously known to put a lot of smoke-and-mirrors without answering a question simply because they are artist in this field and they use the words in such way that they say a lot but they will not answer your question. (Usually they would answer a completely different question where they feel comfortable answering.) 

So that's on a nutshell an intro of why to dodge a question; back to your original question: 

> How should I deal with people who don't answer questions?

If they play the smoke-and-mirror trick on you do what lawyers like to do: **guide them.** Given them the answer and ask them if what you said is true. 

For example:

> Would it be a true statement to say that you're against plan X because of A & B?

Keep it simple so they can digest the question. Ask them if they can start with a YES or NO and then build their argument. 

If your questions are reasonable as you state then you should expect a reasonable answer. Don't allow them to lose focus and tell them that you didn't understand the answer if the question is [here comes the initial **simple** question].