That's wrong and illegal. You should advise your boss that the customer should be contacted and asked for permission to charge the additional amount. Most people would probably understand and agree. That doesn't mean your boss can decide that for himself.
-Alex

I'd go further than Russ and say it definitely would be improper and illegal to charge the card. The sale was complete and unless you were the sales person you cannot know whether or not the error was made at the time of the sale, or whether there even was an error.
In your shoes, difficult as it is to do, I would respectfully decline to perform the requestd transaction. The "call me a fraidy cat, but I'm not comfortable doing this" approach. If your boss thinks it is the right thing to do, he/she can do it. Its your ass awaving in the breeze - cover it.

You are absolutely in the right here, Mizmojo- the business, not the customer, made the error. The sales receipt that the customer was given (showing the amount she paid for the item in question), basically constitutes a contract between the her and the store. The store does not have the right to add charges to the customer's credit card. In fact, it seems pretty dodgy that the business has access to the customer's credit card number after the transaction anyway, unless she has some kind of an account with them, in which case I'd expect that her contact information be available.

There are certain downsides to owning a business, one of which is that when one of your employees makes a mistake you generally have to eat the cost. One example of this is banks. My brother-in-law works as a teller and several months ago a regular customer came in to make a deposit to his business account. The customer had miscalculated his deposit by $1000. My brother-in-law did not catch this error when checking the deposit, and by the time the bank caught the mistake, the customer was gone with a deposit slip showing that he'd made a deposit of $1000 more than he really had. The bank had to eat the loss and supply the extra $1000 to the customer's account.

I would stand up to my boss and just respectfully point out that this is inappropriate and that you are not willing to do it. If you can find the customer's information, contact her, and explain the situation, she may be willing to remedy the situation willingly, but that is the only acceptable way to recoup the loss.

If your customer or her bank has any common sense, they'll charge your boss with fraud when/if they do this. And win.
You aren't uptight at all - you are ethical. and maybe a little bit underinformed by your employer regarding the business you all are in - training should have covered this, in an ideal world.
You should abstain from being involved.

Since it was your company's mistake and the customer did not fraudulently underpay - you cannot "fix" your mistake by charging the visa number for the money. Doing so is theft (perhaps even a case of identity theft) and you could lose your business license.