SMRT doesn't allow you to completely delete a customer because you may want to keep their historical data. However, anyone with *at least* Store Manager credentials can disable an account.
Follow the steps below to disable a customer:
Open the account you want to disable
Click on the customer's name to access their profile
Click on the "Admin" tab
Select "Deactivate Account"
Choose a reason for deactivating the account
Enter a note (optional)
At a first glance it won't seem like anything happened other than a note being added at the top right corner of the account but if you look up the customer again a pop up will open displaying the reason the customer was disabled. After a brief amount of time it will also stop appearing as a search result.