If my entire training set of documents is class A, how can I use TF-IDF to find other documents of class A?
The easiest thing to do is what was already proposed - clustering. More specifically, you extract a single feature vector from set X and then apply K-means clustering to the whole X & Y set. 
ps: Be careful not to confuse k-means with kNN (k-nearest neighbors). You are able to apply only unsupervised learning methods.
I have a collection X of documents, all of which are of class A (the only class in which I'm interested or know anything about). I also have a much larger collection Y of documents that I know nothing about. The documents in X and Y come from the same source and have similar formats and somewhat similar subject matters. I'd like to use the TF-IDF feature vectors of the documents in X to find the documents in Y that are most likely to be of class A.
In the past, I've used TF-IDF feature vectors to build naive Bayes classifiers, but in these situations, my training set X consisted of documents of many classes, and my objective was to classify each document in Y as one of the classes seen in X.
This seems like a different situation. Here, my entire training set has the same class (I have no documents that I know are not of class A), and I'm only interested in determining if documents in Y are or are not of that class.
A classifier seems like the wrong route, but I'm not sure what the best next step is. Is there a different algorithm that can use that TF-IDF matrix to determine the likelihood that a document is of the same class?
FYI, I'm using scikit-learn in Python 2.7, which obviously made computing the TF-IDF matrix of X (and Y) simple.