If by vague prayers, you mean careless, unfocused, unburdened prayers I doubt if they can be answered.
Last week I shared my experience with vague prayers. But we should never base theology on human experience. What does the Bible say about this matter? Well, first of all I need to say that the Bible never says God does not answer vague prayers. However, it does discourage some things that fall into the category of vague prayers. If by vague prayers we mean careless, unfocused, unburdened prayers, I doubt if they can be answered.
And the Bible does encourage specific prayers. Passages like the story of Jesus asking the blind men what they wanted Him to do for them (Mt.20:32) nudge us in that direction. But other passages like God’s pleasure with Solomon’s request for wisdom (2 Chron.1:11) indicate that it is wise to consider even struggle over what we ask for. Specific prayer often needs to be a process. God takes time to hone our vague praying to the sharpened point of His will in a matter. And His will is nearly always different and greater than I would have prayed at first.