I think it is possible but there are many legal connotions with that. For example, security. Most people won't want to enter their credit card details when they do not know how the information is going to be handled, where it will be stored (if at all) and whether or not it is secure.

Most online stores use payment gateways of some form, whether they're custom built or in association with other merchant/payment gateways such as Authorize.net. It's more secure, safer and is much less risky - both for the customer and your client. It's what I'd strongly advise to do.

Usually you'll need an open source shopping cart that works with payment gateways to support your bank account. You can read more here: en.wikipedia.org