The word "prayer" means "a person who prays". See http://www.thefreedictionary.com/prayer, definition 2. (The third block, not #2 under one of the first two blocks.)
Note that when spoken, "prayer" as a person is pronounced pray-er or pray-or, two syllables, while "prayer" as the thing you say is pronounced "prar", one syllable. But in print they are spelled the same. This may be confusing if the sentence is not clearly worded. (If I found it difficult to word the sentence to make the intended meaning obvious, I might write "pray-er" to make it clear. Consider a sentence like, "God blesses both the prayer and the prayer", meaning both the words and the person who says them. Someone reading that without having just read a discussion of the word "prayer" might well say, "Huh?" But if I wrote "... the prayer and the pray-er", they'd probably get the idea.)
In the context you are using, we often say things like "a praying person" or "a person who prays".
If you were talking about someone who stands in front of a church or other group to recite a prayer, he is sometimes called "the prayer leader", even though he's not really "leading" anyone, he's just saying a prayer by himself.