They sort of already have this (IIRC Perplexity also has it), there's a "shopping" specific mode which presumably adds affiliate links. But even GPT-5.1 hallucinates a bit too much to be trusted buying anything without me as a human-in-the-loop. I tried to have it help me find climbing shoes for my cuboid feet and it went from making up specifications to telling me to buy shoes that don't fit and cutting them up with tools which wouldn't work.

EDIT: Looks like they don't add affiliate links? I'm surprised, it seems like a natural thing to do outside of the incentive to bias purchasing sources.