Price.com never asks for your bank login, and you do not need to enter a card number to earn Cash Back on online purchases.
Online Cash Back is tracked through affiliate links when you click through to a store, not by reading your transactions, so your payment details stay with the store you buy from.
In-store card-linked offers do require you to enroll a card. That is handled by Olive, a PCI-compliant card-linking provider that works directly with Visa and Mastercard. Your card number is never stored on Price.com servers.
You can review how your information is handled in the Price.com Privacy Policy.