That's an interesting question. How would the DB handle a request against one of their own? Most likely they may send the unlucky DB member off on an assignment they know they won't survive. This way, the DB member dies and they don't have to violate a tenet.
That assumes they even get the request in the first place. Even if the Night Mother answers a person's prayer, she can probably choose whether or not to pass the request along to a Listener. (Or to wait and pass a request along--say, until the target is no longer a member of the Dark Brotherhood. Assuming she wants that person dead. If she doesn't, tough luck.) And she seems the choosy, biased type.
Basically, praying to the Night Mother in no way guarantees that the person you want dead will actually get the axe, and you'll probably never know why.