Kidlike
The functionality you are asking for does not exist in a default SME server.
Also there are no add on contribs that I am aware of that will give that functionality.
Here is what you could do utilising an existing contrib & standard SME server functionality.
Minimally a new user would send an email request (you could even have a form filler page to do that), then the administrator adds the basic details of username & password, firstname & surname in server manager User panel.
The administrator would install the user manager (smeserver-userpanel contrib), then users can access that to change their password.
Note that the email address is determined by the user name, unless you add pseudonyms.
When the administrator sets up the new user initially, he would also have to enable limited access to user manager.
Exactly how much access is allowed & to what panels is really your decision.
The only issue is that external (remote) user manager access from all locations (public IP's ) has to be allowed, which is potentially a security risk.
Just ensure that user password security is set to highest, I think the SME server default should be OK, so as to prevent users from creating simple easily crackable passwords.
If you just want to provide email, then is webmail sufficient ?
Otherwise you have a significant coding job to do, & you will be tapping into areas of SME server that are then vulnerable from a security viewpoint, & in effect opening a "channel" for hackers to try & gain access to your server backend.
If you go this route, then take a lot of care with the quality of your code.
For a small user base I do not see my suggested method as being cumbersome, but for a large user base you should use a OS system specific to your purpose, ie something a small ISP would use etc. SME is not really the right product.
Read the Pseudonyms section of the Manual for further details of domain based email address issues, & workarounds. This also involves webmail & default return email addresses, so some careful thought is needed before implementing all this (depending on your requirements etc).