Are you going to be taking credit card numbers? I see you plan to email the list of products back to the company. Do NOT email credit card numbers (please think about how you'd like your credit card handled).
If you have a lot of time on your hands, this is definitely do-able. Although I think its fair to say everyone writes funky and buggy code when they first start out. I don't think you'd want your first program to be a shopping cart. I might almost recommend building all the cart components seperately for a different app (or build a cart for a test/demo/personal site for giggles). Figure out the hurdles involved. Then when you have it complete, do a rewrite and think about what worked in the first version and what didn't. Seems kind of extreme, but since a cart HAS to work 100% perfectly on the first day, I don't know if you'd want to risk (or afford) having funky/buggy code for something as high profile as a cart (if it was a forum, guestbook, or something else, then yeah, make it your first project and have fun with it).
You do have alternatives like osCommerce or ZenCart. Although trying to tweak the code may be difficult (they're pretty complex), you would have a working shopping cart right out of the box with some really nice features.