Is there a custom in which a man approaches his Rav to act as a liaison between himself and a lady he is interested in in order to start the dating/courting process? Someone recently recounted to me how she and her husband met. She stated that after someone in the community suggested the match to her now husband, their dating was initiated through the Rabbis. I am trying to ascertain if this is an established custom.
In my (limited) experience, this is a fairly common thing.
Rabbis often meet many different people in the course of their work, and person A might be a woman who has been looking for a match, while person B might be a man looking for a match.
Many derivatives of this exist, such as "person A has brother/sister looking for a match," etc.
In this case, the Rabbi is acting as a shadchan, a match-maker. Generally, most people are qualified to suggest matches (but not every suggestion equals a match), but due to rabbis often being in contact with many more people than the average person in the community, it is sometimes easier for them to suggest potential matches.
Hope that helps.