Yes! You can also change payment information by adding a card or changing a default card on behalf of a client. Learn how here.
If you would like to manually charge a customer on a one-off basis, you can place a New Order on the roster of the class they are planning to attend.
Customers enter their payment information when they sign up for the first time on Sawyer. The information is encrypted so nobody can view it. Customers can also update their payment information in their Sawyer account.