I have a customer, wants me to build the shopping cart, but he wants to be able to handle the backend day-to-day product adding, order fulfillment, stock adjustment, etc.

Can we purchase one SCCP and have it installed on his computer to do what he needs and my computer to do my part?

I'm pretty sure you'll need two separate licenses because it's two different users.

Yep just to add agreement here, Eric is right. You will each need your own license to do this.

Take advantage of https://www.coffeecup.com/store/pricing/

And get the customer to open their own CC account buy it through their account, and give you access so you get the second copy cheaper and paid for by them, its tax deductible for them

You can also buy it on specials for them and have the copy transferred to their account, so this way when there are sales you can opt to save them some money too. Lots of ways to acquire the software from CC so it shouldn't be too difficult to save some cash