Implement Shopping Cart (Woo, or similar) to an already functioning website

Implement Shopping Cart (Woo, or similar) to an already functioning WP website, however no payments, SSL or financial transaction is required. All we want is to add items to a cart, so it adds them all up, adds tax, etc. So the client can see what the total cost will be, when they click 'confirm quote' it emails us and them with the items added up, tax added, etc. That is all, we will invoice them manually, so no payment handling is required. This is our web site [login to view URL] which is serving us very well, however our clients are wanting to hire more and more individual items and this is making it difficult for them because there is no cart function, they have to write them all down and then type them out in the contact us form, then we have to add them all together and then email them with a quote, it wastes a lot of time. Historically it didn't matter because our clients would hire one or two items, but frequently they want quotes on several hire days with several items, etc. A cart function that would add the number of items/costs and the number of hire/chargeable days would give them an idea of costings straight away and save us the time typing it up in a PDF every time. The end of the process is them clicking 'confirm quote' and we will contact them after. No payments are required, just an email to us and them with the items, total figure and tax. If all of the items on our site need to be added to the shopping cart system then we can do that, so you don't need to do any data entry, just do a couple of samples and show us how and we will do the rest. Our site is using a custom WordPress theme, not an off the shelf theme.