How can a company issue a visa or mastercard logo debit card without
using an bank as an issuer?  The idea is to allow clients to have a
rechargable debit card that can be used for card-not-present
transactions.

My university ID card is a smart card, which stores a limited amount
of money on the card that I can use anywhere affiliated with the
university. Although it doesn't have a VISA or Mastercard logo on it,
it as well, is not endorsed by a bank. There is a central database on
campus that no matter where you use the card, information is sent to
the computer and the specified amount of money is taken off the card.
When you run low on funds the cashier informs you of your balance and
you can add money at the central office. In function these smart cards
act like a debit card, but no bank is required for them to work.

I don't know of any debit card companies that issue with Banks.  In
order for you to have a Bank issue you a debit card you will have to
have an account with them.  I really don't understand why you want a
bank to back up the debit card.  The idea behind the debit card is to
keep you from carrying large amounts of cash on your person.  Thus
keeping your money safe.
The short of this is very simple: The USA PATRIOT Act is a Federal law
that requires all financial institutions to obtain, verify, and record
information that identifies each person who opens an account Banks or
Otherwise. They will ask you to provide your name, address, date of
birth, and other information that will allow them to identify you. You
may also be asked to provide them with documentation as proof of
identification. Please be aware that "100% Approval" and "No one is
Turned Down" is contingent upon successfully passing their mandatory
identification confirmation. So if you are thinking of a debit card as
a way to hide money or something along that line. It won't work.
On the other hand if you are looking for a Debit Credit Card then I
suggest you apply for one at
http://www.creditcardcite.com/debit_credit_card.htm  this is where I
obtained the one I'm using now.  There are many for you to choose from
with many different issuers.

There are a great number of debit card companies that issue with
Banks.  There are also a lot of debit card companies that don't issue
with banks that have the Visa and Mastercard logo. In order for you to
have a Bank issue you a debit card you will have to have an account
with them.  I really don't understand why you want a bank to back up
the debit card.  The idea behind the debit card is to keep you from
carrying large amounts of cash on your person.  Thus keeping your
money safe. No matter what the logo or issuser. If you have an
underlying question that you want answered in a round about way read
more.
The short of this is very simple: The USA PATRIOT Act is a Federal law
that requires all financial institutions to obtain, verify, and record
information that identifies each person who opens an account Banks or
Otherwise. They will ask you to provide your name, address, date of
birth, and other information that will allow them to identify you. You
may also be asked to provide them with documentation as proof of
identification. Please be aware that "100% Approval" and "No one is
Turned Down" is contingent upon successfully passing their mandatory
identification confirmation. So if you are thinking of a debit card as
a way to hide money or something along that line. It won't work.
On the other hand if you are looking for a Debit Credit Card with the
Visa or Mastercard log not associated with a bank - then I suggest you
apply for one at http://www.creditcardcite.com/debit_credit_card.htm 
this is where I obtained the one I'm using now. No Bank is involved. 
There are many for you to choose from with many different issuers and
terms so you can be picky...