OK, narrowed it down to Internet Explorer Cookies.
If they have their security setting set to high in MIE, they cannot add anything to the shopping cart.
There are no warnings of any kind given about cookies. It just does nothing when they click on Add to Cart. I replicated this on my own computer with Explorer with the security slider set to Medium. It would not allow me to add anything to the cart.
If I added my web site URL to the list and set it to "allow cookies" then it works. Or if I set it to "Low" on the slider it works.
So, does every potential customer have to go through this process just to buy something? If so they are going to leave and go somewhere else. #1 they are not going to have any clue why they can't add anything to the cart and #2 They aren't going to go through a tutorial on how to change the settings for cookies.
Net result: Lost sale.