I can't recommend a particular program from experience, but a quick Google Search turned up numerous hits. Maybe check some of these lists and see what program has the features you need.

Zen Cart looks like the most popular one on Sourceforge. NopCommerce on Codeplex seems popular. SimpleCart and OpenCart on Github are popular.

Oh, and if you find a program which is at least close to what you want but not exactly, then file a feature request or contact the author(s) directly. You might be able to have the needed functionality addd.

Search sorceforge.net

I need a shopping cart that will either accept an extended (with many custom fields) registration OR ask the customer for several fields of information after they click the item to add it to their cart, in order to accept orders for highly customized product. The kicker is I don't have the patience or skills of a programmer so I need something with an intuitive UI. I've been researching for days and haven't found anything that would work without extensive changes to the code. Is there any such animal or am I going to have to pay for someone to write something for me?