Please tell me why you don’t eat meat.
どうぞ in this sentence sounds quite unnatural. どうぞ～下さい means “please go ahead and do X”, “please feel free to do X”, or “please do X if it doesn’t bother you”.
But the original sentence suggests that the speaker would like to politely ask people to tell the reason. In this case, we don’t say どうぞ～下さい. I would propose:
あなた方が肉を食べない理由を教えてください。(i.e. omitting どうぞ still sounds polite because of 下さい (please). This is the closest to the original English sentence.)
あなた方が肉を食べない理由をどうか教えてください。(i.e. inserting どうか, not どうぞ, means “I am begging you to do X”. The demand is strong yet very polite.)
あなた方が肉を食べない理由を教えてくださいませんか？ (Another polite expression. This is closer to “would you mind if I ask you the reason why you don’t eat meat?”)