My own experience--nearly all observation, not what happened to me--is that the unknown playwright submits full-length plays to theatres which put out calls for new works or contest notifications. Most playwrights start locally.
I live near a smallish city (< 1 million) and there are multiple places to submit plays, although the equity-level theatre is clearly head and shoulders above the others in terms of prestige. Their websites and the local paper put out the calls for submission and the requirements (often for length and/or number of players).
A play I saw for free as a winner of a contest at the equity theater, produced in a script-in-hand minimalist play, later saw full production in a few theatres well outside the region, then was purchased by the theatre which first saw it, with normal production values. It was pretty cool to note what had changed.
Maryn, whose one-act also won, which is why she was even there to see the other play