Is it possible to set up some kind of system in my company whereby a
customer can swipe his/her debit card, and have my company credit that
person's checking account with a specific amount?  I want to be able
to credit customer's accounts without any previous transaction (i.e.
not just as a credit for an exchanged item).

Check the Terms of Service with your Cedit Card Terminal Provider. 
Retail store normally will not offer that type of transaction due to
high potential for Fraud.

My company termial it is avialable just processing as a credit instead
of a sale, but we don't use it, because of the fraud aspect and
potential Problems resulting from "terminal provider" - (EMS.

So the simple answer would be look for an alternative.  Too much room
for Fraud, and the Credit Card companyies themselves "red flag" and
investigate those types of transactions.