This is a possible approach to solving your problem (with more knowledge about your website and your requirements, approaches with more elaboration will be conceived):
Edit the main template for the site (if it has one) so that the navigation bar has a shopping cart icon that links to a new, internal shopping cart page.
Make it so that former buttons that linked to the separate shopping cart site now go to the new internal shopping cart page.
As well, add function where users can add items to the shopping cart and previously mentioned shopping cart icon indicates addition with an animation showing a number representing updated amount of items in the shopping cart.
Shopping Cart page -
Lists products added to the cart in a table. Includes columns for image of product, cost, vendor, more details
Has functions for removing a product from cart
Function for changing the # of a particular product they want to order
Checkout button
Checkout page -
Selecting method of payment (PayPal, credit/debit card, etc)
Select shipping & billing addresses
Back/Return button
Order button
Confirmation page:
Lists relevant details of the processed order
Button to return home for more shopping
Questions:
Does this website have function where users sign in and have a user account? Relevant for if user signs in after already adding items to their cart,
What kind of website is this and what language is it programmed in? Is it a WordPress site, a custom PHP site?