Erase User

I have been advised that I need to offer a delete account option/erase their own data option from the user side.

I have advised of the current process we handle under gdpr and how we then processes that request.

If there a way a user can select erase data or delete account from under there account settings or is it only something I can do.  

Only you can do it (People -> Erase People Search). 

You could perhaps create a form and contact page for them to request and link it in the footer.