Is it obligatory to pay a commission to a website that requests 1% of the sale value, given that the buyer does not mention the source of the advertisement, and communication with the website is unavailable to confirm their entitlement?

Unless there is proof that the buyer knew about the item through the website, you are not obligated to pay him anything; because the default is the exoneration of liability, and rights are not established merely by doubt. It is worth noting that the website's condition of taking a percentage of the sale price in exchange for advertising and promotion is a matter of dispute among scholars.