Take a look at this extension called **Delete My Account**:

http://extensions.joomla.org/extensions/clients-a-communities/user-management/26011

It allows users to delete their own account via the frontend. There is also a feature where it notifies all admins if a user has deleted their account. Looks like a promising extension and exactly what you need. Compatible with Joomla 2.5 and 3.x

**Update:**

I have just tested this plugin on my localhost and it works perfectly fine.

 1. Install the plugin
 2. Enable it and in the options, set **Action to be taken** to **delete**
 3. You then need to place a blank with wit a specific class. You can place this link literally anywhere you want. I decided to put it in an article, simply to test. So add the following somewhere on your site: `<a href="#" class="f90-delete-my-account">Delete my account</a>`
 4. When you click on this link in the frontend, it should load a modal box asking you to confirm that you want to delete your account.