Here is a sample template you can use to write a bank authorization letter.
The Branch Manager
I am writing this letter to inform you that I am currently a [savings/credit] account holder at your bank. My account number is [account number] and due to inevitable conditions, I am unable to visit the bank for [concerned activity]. This is why I hereby authorize my [son/wife/etc.], [Name] to operate my bank account on my behalf.
I request you to cooperate with the same and allow [him/her] to make the operations. I have provided him/her with the account details and documents, and he/she will provide identity proof for verification.
For any further clarification, kindly reach me out at [phone no.].