Why are you looking for a BAPI? It looks like you want to control the ability to change the email so that only select group of users can make the change?
If so, Please explore the t-code SHD0. You can create transaction variants using which you can control the exact data that you would like for a user group to be able to edit.
FYI you can look at the BAPI-
BAPI_CUSTOMER_EDIT if you want to use that to modify the email.
Hope this helps,
Ravin