What you are asking for is technically achievable but will require customizations. It is by design that Shopping Cart relies on cookie and cache in the session because it works well for most e-commerce web scenarios. But shopping cart behavior is customizable, please refer to the documentation.
For example, you can have your own custom implementation of ICurrentShoppingCartService which will be reading cart GUID from Web API request parameters rather than cookies. And in the code you can use ShoppingCartInfoProvider with ShoppingCartInfo objects. I hope it gives you an idea how to proceed with this.