Does that include the Delete Account button/feature?
Already tested this on my own board using a dummy account and making one post, then deleting the account as dummy member. Their posts get left behind using same name they registered with. So it's not that bad really, it's not as though all their posts go with the account, or get listed under GUEST name (once their account is deleted).
I don't see that member being able to delete their own account as a bad thing, saves them pestering you to delete it if they want to do it. Let them do it themselves, but all their posts still remain intact. Which is all that matters really for the sake of keeping threads making sense they posted in.