I would use a dynamic reader (ie one with just a single source feature type with the merge filter set). Then expose the fme_feature_type format attribute. This will give you the name of the table being read.
You can run the data through a DuplicateRemover too, removing duplicate values for fme_feature_type, in order to get a list of unique table names.
There is a Schema format that returns the schema of a dataset, but I think it works only with file-based data, and wouldn't work with a File Geodatabase (though it might with a Personal Geodatabase).