HTML code for adding shopping cart and redirect traffic possible?

If I have affiliate links, is there any chance to add them to a shopping cart on my website and when they are ready to "Cash Out", I send them to the sites shop with the cart and the added items?

Amazon has it and I wonder if It is possible to do the same to other sites?

They explain it probably better:
For the greatest amount of control and influence over your visitors' shopping experience, use Product Advertising API to build a remote ****** Shopping Cart on your site. As they browse your site, your visitors will be able to add products to their cart without having to leave the page. When they're finally ready to complete their order, the contents of their cart will be available at ****** for secure checkout.

Thankful for any answers!

It all depends on the site they checkout at and if they support it.
But with only HMTL, no. Most likely it's with help of a cookie.
There's plenty of amazon wordpress themes like this if you don't want to hire someone to create the solution for you.