Host your product on another website where it'll be included a customer's checkout cart/balance?

[AustralianDream]

Hi,

I sell Printfull dropship products on my Shopify store and would like to host those products on client websites so that whenever a customer purchases something on a customer's website, my product would be included with the client's different products into 1 complete checkout cart for the customer. Is this possible and if so, how? If its not possible, how much would it cost to build/set up this functionality?

Note that the client sells a mixture of their products and mine and we don't want the customer to move websites as part of their shopping experience, they would stay on the client's site only.