Hi Ty,
Unfortunately, I am not familiar with a way to specifically call for a search widget from an outside website. We do have our API Guide which does go over running API calls such as Do Query that a developer can embed into an external website to call into Quick Base and do specific searches.
https://help.quickbase.com/api-guide/index.htmlOur APIs allow for external services to communicate with Quick Base, they just take some development to hone in on the needed refinements. We do have Partners that offer their services to do external integrations with Quick Base.