The new responses API has web search capability; however, when used, it currently searches broadly and returns results from any domain. Is there a way to restrict the search or limit responses to specific domains?
For instance, if a user asks, “I am looking to buy laptop bags,” can the search results be limited exclusively to suggestions from myecommerce.com and mybags.com? Is this functionality possible?
I just tried this as a “hack,” and it seems to work?
search site:litrpgreads.com
works but doesn’t return the exact source url from which it got the answer
As of now it’s not possible to restrict the search source domains using request config.
I’ve forwarded this feature request to the OpenAI team. 
In the meantime,@PaulBellow ’s approach with prompting is definitely worth a shot as it can be used to filter the final output - especially for blocking certain domains.