is it possible to have this setup to take a credit card payment through my own merchant account and banking institition?
No, CBSubs uses Payment Gateways. Currently the only supported gateways are as follows.
Paypal
Authorize.net
YellowPay
Offline
Dummy (for testing)
With next release we will be providing the following additional gateways.
Ogone
Swiss PostFinance
2Checkout
We will then later be providing support for many more gateways, but direct payment in the way a local retail store would function is not provided. If you want on-site purchases I recommend Authorize.net (you will need a dedicated IP address and valid SSL certificate).
It might be possible to integrate direct payment to your bank once the CBSubs API has been provided (later down the road, but planned).