Is it possible that science be used to demonstrate that prayer actually works? No. It cannot. Here's why.
Science is the process of learning about the universe and how it behaves. The scientific method deals with observation, developing a hypothesis, testing the hypothesis through experimentation, and then developing a workable theory to explain behavior.
So, we have to ask how science could be used to demonstrate whether not prayer works. Do they want repeatable experiments and regular quantifiable data so that the efficacy of prayer can be tested and measured? That would be a problem. If prayer "A" resulted in effect "B," then we would see a correspondence of prayer and result -- something the atheist could see and verify. But if this were the case, such a phenomena would not be a demonstration that God exists. Instead, it would be a demonstration that uttering certain words in certain patterns brings certain results. This would imply that a new property of the universe has been discovered, and that by saying certain words certain results occur. Essentially this would amount to sorcery and would not demonstrate that God exists.
Furthermore, prayer to God is prayer to living being. He is not in the habit of being a butler who jumps at every request made to him. He is a living being who is the sovereign Creator of the universe and works things according to his will, not hours. Requiring God to submit to testability in the issue of prayer is equivalent to requiring him to submit to our whims and narrow sighted wants. This is often the behavior of children who pout when they don't get what they want and then proclaim that the parents don't really love them. Those who challenge God of the scientific basis and see prayer is not produced the wanted results, and then complain about God saying he doesn't exist, are similar to those children who don't get what they want when your prayers aren't answered the way they want. Then, in their arrogance they confidently boast to others that God does not exist. All the while, they fail to realize that they cannot subject God to their demands.